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COURSE  OF  EXPERIMENTS 


PHYSICAL  MEASUREMENT, 


En  Jour 


PART  I. 

DENSITY,  HEAT,  LIGHT,  AND  SOUND. 

3  y  b  i. 
BY  HAROLD  WHITING,  PH.D., 

FORMERLY  INSTRUCTOR  IN  PHYSICS  AT   HARVARD  UNIVERSITY. 


SECOND   EDITION. 


BOSTON,  U.S.A.: 
D.  C.  HEATH  AND  CO.,  PUBLISHERS. 

1892. 


98 


Copyright,  1890, 
BY  HAROLD  WHITING. 


SSmbrtsttg  $ress: 
JOHN  WILSON  AND  SON,  CAMBRIDGE,  U.S.A. 


QC31 


PREFACE. 


THIS  book  is  intended  to  aid  in  the  preparation 
of  students  for  courses  in  civil,  electrical,  or  me- 
chanical engineering,  and  for  advanced  work  in  all 
branches  of  science  requiring  the  use  of  accurate 
methods  and  instruments  of  precision.  To  this  end, 
the  course  of  experiments  described,  unlike  that  con- 
tained in  most  manuals  published  in  this  country,  is 
exclusively  devoted  to  quantitative  physical  determi- 
nations. Comparatively  little  use  is  made  of  the  or- 
dinary experimental  demonstrations  of  well-known 
physical  laws  and  principles,  which,  it  is  believed,  are 
better  suited  to  the  lecture-room  than  to  the  labora- 
tory. Most  of  the  experiments  consist  in  the  deter- 
mination of  magnitudes  wholly  unknown  to  the 
students,  and  are  made  with  instruments  which  they 
themselves  have  tested,  in  order  that  they  may  learn 
to  depend  upon  their  own  observations. 

Attention  has  been  paid  throughout  this  book  to  the 
general  methods  which  underlie  all  physical  measure- 
ment, rather  than  to  the  special  devices  by  which 
particular  difficulties  are  overcome.  It  is  considered 
of  greater  advantage  to  show  how  comparatively 


vj  PREFACE. 

accurate  measurements  may  be  made  with  rough  ap- 
paratus, than  to  explain  the  use  of  instruments  of 
precision  which,  in  the  hands  of  students,  are  apt  to 
give  erroneous  results.  The  apparatus  required  for 
this  course  is,  accordingly,  of  the  simplest  possible 
description. 

Most  institutions  are  obliged,  by  considerations  of 
expense,  to  limit  either  the  quantity  or  the  quality  of 
the  instruments  provided  for  the  laboratory.  When 
the  supply  of  apparatus  is  insufficient,  the  work  of  a 
given  student  at  a  given  point  of  time  is  obviously 
determined,  to  a  greater  or  less  extent,  by  the  instru- 
ments which  happen  to  be  free  for  him  to  employ,  and 
the  systematic  instruction  of  large  classes  becomes 
impracticable.  This  book  is  intended  especially  for 
use  in  laboratories  which  are  or  can  be  provided  with 
a  liberal  supply  of  moderately  accurate  apparatus. 
Effort  has  been  made  to  devise  inexpensive  instru- 
ments, especially  when  several  copies  of  a  given  kind 
are  likely  to  be  needed ;  and  it  has  been  found  that, 
notwithstanding  the  expense  of  all  necessary  redupli- 
cations, a  considerable  saving  may  be  effected  by  the 
"  Collective  System  "  of  instruction  in  the  cost  and 
labor  of  conducting  an  elementary  laboratory  course. 
The  experiments  are  accordingly  such  as  can  be  once 
for  all  explained  to,  and  within  a  reasonable  length  of 
time  performed  by  a  large  class  of  students.  They 
are  moreover  arranged  in  a  connected  and  progressive 
order. 

The  care  and  accuracy  required  to  obtain  concord- 
ant results  in  Physical  Measurement,  the  continual 


PREFACE.  Vli 

use  of  experimental, inductive,  and  controlled  methods, 
give  to  that  science  a  peculiar  educational  value,  aside 
from  the  natural  laws  and  principles  with  which  the 
student  must  become  familiar.  The  course  of  experi- 
ments has  been  adapted,  in  so  far  as  possible,  to  the 
needs  of  students  who,  having  little  or  no  previous 
training  either  in  mathematics  or  in  physics,  wish  to 
obtain  a  general  scientific  education.  Every  branch 
of  physics  is  accordingly  represented  by  typical  ex- 
amples. In  order,  however,  not  to  exceed  the  natural 
bounds  of  an  elementary  treatise,  the  author  has 
limited  his  selection  to  such  experiments  as  have  been 
proved,  practically,  in  his  own  experience,  to  yield  the 
most  satisfactory  results  from  an  educational  point  of 
view.  It  is  hardly  necessary  to  add  that  these  experi- 
ments involve  physical  measurement  in  every  case. 

The  amount  of  mathematics  required  in  the  use  of 
this  book  is  not  so  great  as  might  be  supposed  from 
a  casual  examination  of  its  pages,  since  many  proofs 
are  given  in  full  which  in  other  text-books  are  taken 
for  granted.  The  course  of  one  hundred  experi- 
ments involves  only  the  simplest  propositions  in  arith- 
metic and  geometry,  and  little  or  nothing  of  algebra 
or  trigonometry  beyond  the  mere  notation.  Prob- 
lems presenting  any  special  difficulty  are  treated 
separately  in  a  portion  of  the  Appendix  (Part  IV.) 
not  intended  for  general  use. 

The  first  part  of  this  book  relates  especially  to  hy- 
drostatics, thermics,  optics,  and  acoustics ;  containing 
measurements  of  mass,  density,  length,  temperature, 
heat,  light,  and  wave-lengths  of  sound. 


viii  PREFACE. 

The  second  part  contains  all  such  measurements  as 
involve  motion  or  acceleration.  That  part  of  acous- 
tics which  relates  to  the  measurement  of  time  is 
also  included  ;  then  follow  dynamics,  magnetism,  and 
a  comparatively  extended  series  of  electrical  meas- 
urements. A  few  experiments  intended  (with  cer- 
tain exceptions)  for  advanced  students  are  added, 
together  with  a  description  of  certain  instruments  of 
precision. 

The  third  part  contains  notes  on  the  general 
methods  of  physical  measurement,  and  on  physi- 
cal laws  and  principles.  An  extended  series  of 
mathematical  and  physical  tables  is  also  included  in 
this  part. 

The  fourth  part,  or  Appendix,  contains  suggestions 
to  teachers  in  regard  to  laboratory  equipment,  appa- 
ratus, expenses,  and  methods  of  instruction.  It  in- 
cludes a  full  set  of  examples,  showing  how  the 
observations  in  the  course  of  one  hundred  experi- 
ments should  be  recorded  and  reduced.  These 
examples  embody  results  a  great  part  of  which 
were  actually  reported  by  students.  There  are  also 
three  working  lists  of  experiments,  of  different  lengths 
and  degrees  of  difficulty,  and  proofs  of  certain  im- 
portant mathematical  formulae. 

The  text  of  the  first  and  second  parts  is  divided 
into  short  chapters,  distinguished  by  the  names  of 
the  experiments  (Exps.  1-100)  to  which  they  relate. 
The  experiments  are  still  farther  divided  into  sec- 
tions (^|^|  1-270),  devoted  in  some  cases  to  the  practi- 
cal, in  other  cases  to  the  theoretical  treatment  of  the 


PREFACE.  IX 

subject.  It  has  not  been  thought  necessary  or  desira- 
ble to  indicate  in  all  cases  just  what  portions  of  an 
experiment  the  student  is  expected  to  perform,  and 
what  portions  it  is  sufficient  for  him  to  read.  This 
must,  of  course,  depend  largely  upon  circumstances. 
Full  directions  for  each  of  the  one  hundred  regular 
experiments,  or  for  each  part  of  which  it  consists, 
will  usually  be  found  in  a  separate  section  headed  by 
the  word  "  Determination."  In  the  case,  however,  of 
outside  experiments  mentioned  only  for  the  sake  of 
illustration  or  continuity,  directions  are  either  entirely 
omitted,  or  replaced  by  a  mere  outline  of  the  meth- 
ods involved,  with  which  it  is  important  that  the 
student  should  become  acquainted.  Examples  will 
be  found  under  the  "Peculiar  Devices  employed  in 
Calorimetry"  fl[  97),  and  the  "Velocity  of  Light" 
(^[  247),  which,  though  obviously  impracticable,  even 
for  advanced  students,  furnish  reading  matter  which 
is  none  the  less  instructive. 

More  than  half  of  the  sections  in  the  first  and 
second  parts  relate  to  principles  involved  in  the  ex- 
periments, the  construction  of  the  necessary  appara- 
tus, or  the  calculation  of  results.  These  should  be 
read  or  omitted  by  the  student  at  the  discretion  of  the 
teacher.  The  references  to  the  third  part  (§§  1-156), 
which  occur  throughout  the  experiments,  should  be 
looked  up  by  the  student  in  the  order  in  which  they 
are  met,  and  afterward  read  consecutively.  The 
teacher  should  make  sure  that  these  references  are 
understood,  in  the  case  especially  of  students  who 
may  have  had  no  previous  training  in  physics. 


x  PREFACE. 

The  examples  in  the  fourth  part  are  intended  to 
aid  the  teacher  in  preparing  a  list  of  the  data  re- 
quired for  a  given  determination,  and  in  explaining 
the  reduction  of  these  data.  The  calculations  are 
made,  for  the  most  part,  by  purely  arithmetical  pro- 
cesses, and  in  so  far  as  possible,  by  one  step  at  a 
time,  so  that  the  student  can  hardly  fail  to  under- 
stand them.  The  author  has  found  in  his  own  ex- 
perience that  such  examples  can  be  safely  trusted  in 
the  hands  of  students ;  but,  for  obvious  reasons,  it 
was  thought  better  that  they  should  be  contained  in 
the  fourth  part  or  Appendix,  copies  of  which,  sep- 
arately bound,  can  be  used  by  teachers  who  prefer  to 
keep  the  examples  at  certain,  or  at  all  times,  in  their 
own  hands. 

The  three  lists  of  experiments,  proposed  by  the 
author  with  a  view  of  preparing  students  for  various 
requirements  of  Harvard  College,  may  be  useful  also 
to  teachers  who  wish  merely  to  shorten  the  course  of 
experiments  described  in  this  book,  without  interrupt- 
ing the  continuity  of  the  course. 

The  mathematical  portions  of  the  Appendix  contain 
proofs  which  may  be  of  interest  to  ambitious  students 
and  a  convenience  to  teachers  who  find  it  desirable  to 
step  leyond  the  limits  of  this  book. 

Few  references  are  given  to  works  of  other  authors. 
It  has  been  thought  better  in  an  elementary  book  to 
incorporate  in  the  text  such  abstracts  from  the  best 
authorities  as  it  may  be  necessary  for  the  student  to 
refer  to.  The  course  of  experiments  here  described 
was  elaborated  from  one  previously  given  by  Pro- 


PREFACE.  XI 

fessor  Trowbridge, and  outlined  in  his  "New  Physics" 
(Appleton,  188-i).  In  this  course  frequent  reference 
was  made  to  the  well  known  works  of  Everett,  Kohl- 
rausch,  and  Pickering.  It  is  impossible  to  say  to  what 
extent  the  author  may  be  indebted  to  these  sources  for 
the  ideas  contained  in  this  book. 

The  advanced  sheets  of  a  "  Syllabus "  of  experi- 
ments arranged  by  the  author  were  distributed  to 
his  class  in  the  year  1884-1885,  before  the  works  of 
Glazebrook  and  Shaw,  and  Stewart  and  Gee,  could  be 
obtained.  While  considerable  assistance  was  derived 
from  these  works  in  the  preparation  of  this  book,  the 
"  Syllabus  "  mentioned  above  was  taken  as  the  basis 
for  most  of  the  experiments.  The  notes  contained  in 
the  third  part  were  first  distributed  to  students  in 
1888-1889,  but  largely  rewritten  in  1890.  The  tables 
were  condensed,  by  permission,  from  those  of  Pro- 
fessors Landolt  and  Bornstein,  and  from  other  sources 
elsewhere  acknowledged.  The  first  part  was  printed 
in  1890  ;  the  remaining  three  parts  in  1891.  In  the 
same  year  a  corrected  edition  of  the  first  three  parts 
was  prepared  for  the  use  of  students,  and  all  four 
parts  were  combined  in  a  single  volume  for  the  use  of 
teachers  and  students. 

The  author  is  indebted  to  Professor  Trowbridge 
for  an  outline  of  many  successful  experiments ;  to 
Professor  Hall  for  a  revision  of  a  part  of  the 
proof-sheets,  for  numerous  useful  and  practical  sug- 
gestions, and  for  parts  of  experiments  taken  from 
his  elementary  course  ;  to  the  late  Mr.  Forbes,  of  the 
Roxbury  Latin  School,  for  important  criticisms ;  and 


Xii  PREFACE. 

to  Mr.  Edgar  Buckingham,  Assistant  in  the  Jefferson 
Physical  Laboratory  of  Harvard  University,  for  valu- 
able aid  in  preparing  the  course  of  experiments. 

The  author  wishes  also  to  acknowledge  several 
errata  kinuly  pointed  out  to  him  in  earlier  copies,  and 
to  state  that  he  will  gladly  receive  from  any  source 
further  corrections  or  criticisms  which  may  be  of 
service  in  preparing  a  revised  edition  of  this  book. 

CAMBRIDGE,  November,  1891. 


TABLE   OF  CONTENTS. 


$art  JFtrst 

MEASUREMENTS  RELATING  TO 
DENSITY,  HEAT,  LIGHT,  AND   SOUND. 


PRELIMINARY     EXPERIMENTS. 

PACK 

I.   DIRECT  MEASUREMENT  OF  DENSITY 1 

Nicholson's  Hydrometer, 

II.   TESTING  A  HYDROMETER 8 

III.  WEIGHING  WITH  A  HYDROMETER 13 

IV.  WEIGHING  IN  WATER  WITH  A  HYDROMETER   .     .  14 
V.    ATMOSPHERIC  DENSITY 17 

The  Balance. 

VI.    TESTING  A  BALANCE 27 

VII.    CORRECTION  OF  WEIGHTS 38 

VIII.   WEIGHING  WITH  A  BALANCE 42 


xiv  TABLE  OF  CONTENTS. 

DENSITY. 

DENSITY    OF    SOLIDS. 

PAGE 

IX.   THE  HYDROSTATIC  BALANCE,  1 43 

X.   THE  HYDROSTATIC  BALANCE,  II.      .....     46 

The  Specific  Gravity  Bottle. 

XI.   CAPACITY  OF  VESSELS 49 

XH.   DISPLACEMENT,  I ,     .     .     53 

XIII.  DISPLACEMENT,  II. 56 

DENSITY    OF    LIQUIDS. 

XIV.  DENSITY  OF  LIQUIDS 58 

XV.    THE  DENSIMETER  .     .     . 59 

XVI.   BALANCING  COLUMNS 63 

DENSITY    OF    GASES. 

XVII.   DENSITY  OF  AIR 67 

XVIII.   DENSITY  OF  GASES .69 


LENGTH. 

XIX.   MEASUREMENT  OF  LENGTH 71 

XX.   TESTING  A  SPHEROMETER 83 

XXI.   CURVATURE  OF  SURFACES.  .  88 


TABLE  OF  CONTENTS.  XV 

HEAT. 

COEFFICIENT    OF    EXPANSION. 

PAGE 

XXII.    EXPANSION  OF  SOLIDS 90 

XXIII.  EXPANSION  OF  LIQUIDS,  I .     9-t 

XXIV.  EXPANSION  OF  LIQUIDS,  II 101 

TEMPERATURE. 

XXV.    THE  MERCURIAL  THERMOMETER     ....  104 

XXVI.    THE  AIR  THERMOMETER,  L 119 

XXVII.    THE  AIR  THERMOMETER,  II 127 

CHANGE    OF     PHYSICAL    CONDITION. 

XXVIII.    PRESSURE  OF  VAPORS,  1 132 

XXIX.    PRESSURE  OF  VAPORS,  II 135 

XXX.    BOILING  AND  MELTING  POINTS 140 

CALORIMETRY. 

XXXI.  METHOD  OF  COOLING 144 

XXXII.  THERMAL  CAPACITY 157 

XXXIII.  SPECIFIC  HEAT  OF  SOLIDS 178 

XXXIV.  SPECIFIC  HEAT  OF  LIQUIDS 184 

Latent  Heat. 

XXXV.  LATENT  HEAT  OF  SOLUTION 194 

XXXVI.  LATENT  HEAT  OF  LIQUEFACTION    ....  199 

XXXVII.  LATENT  HEAT  OF  VAPORIZATION   .     .     .     .202 

XXXVIII.  HEAT  OF  COMBINATION  .  205 


TABLE   OF  CONTENTS. 


RADIATION. 

PAGE 

XXXIX.   RADIATION  OF  HEAT 212 


LIGHT. 

XL.  PHOTOMETRY 222 

FOCAL     LENGTHS. 

XLI.   PRINCIPAL  Foci 230 

XLIT.   CONJUGATE  Foci 236 

XLIII.    VIRTUAL  Foci 239 

OPTICAL     ANGLES. 

XLIV.    THE  SEXTANT 244 

XLV.  PRISM  ANGLES 255 

XLVI.    ANGLES  OF  REFRACTION 257 

XL VII.   WAVE-LENGTHS     .  .  264 


SOUND. 

XLVIII.   INTERFERENCE  OF  SOUND  .......  270 

XLIX:   RESONANCE 272 

L.  MUSICAL  INTERVALS 273 


PHYSICAL    MEASUREMENT. 


JFirst. 


MEASUREMENTS    RELATING  TO    DEN- 
SITY, HEAT,  LIGHT,  AND  SOUND. 

EXPERIMENT    I. 

MEASUREMENT    OF    DENSITY. 

^[1.  The  Density  of  a  Rectangular  Block. — The 
volume  of  a  rectangular  block  may  be  defined  as  the 
product  of  its  length,  its  breadth,  and  its  thickness. 
If,  according!}7,  each  cf  its  three  dimensions  has  been 
measured  (§  1)  in  centimetres  (§  5),  we  may  find  the 
volume  of  the  block  in  cubic  centimetres  by  mul- 
tiplying these  three  dimensions  together.  When  two 
blocks  are  of  exactly  the  same  size,  but  of  unequal 
weight,  as  for  instance  a  block  of  wood  and  a  block 
of  metal,  they  are  said  to  differ  in  respect  to  density. 
Obviously,  to  determine  the  density  of  a  body,  we 
must  find  its  weight  as  well  as  its  volume.  For  con- 
venience in  calculation,  the  weighing  should  be  made 
in  grams  (§  6),  since  density  is  customarily  expressed 
in  grams  per  cubic  centimetre  (§  9).  To  calculate 
the  density  of  a  body,  we  divide  its  weight  in  grams 
by  the  number  of  cubic  centimetres  contained  in  its 
volume,  and  thus  find  the  weight  of  one  cubic  cen- 
i 


2  MEASUREMENT   OF  DENSITY.  [E.\p.  1. 

timetre.  This  is  the  density  (or  average  density)  in 
question,  expressed  in  absolute  units  of  the  C.G.S. 
system  (§  8).  It  should  be  noted  that  in  this  system 
the  density  of  a  body  is  equal  to  the  weight  in  grams 
of  a  cubic  centimetre  of  the  substance  of  which  it  is 
composed. 

The  density  of  a  fluid  cannot,  for  obvious  reasons, 
be  determined  like  that  of  a  solid,  by  direct  measure- 
ments of  its  weight  and  linear  dimensions  ;  but  when 
the  volume  of  a  block  has  been  found,  there  are  vari- 
ous methods  by  which  the  weight  of  an  equal  bulk  of 
a  fluid  may  be  determined.  We  may,  for  instance, 
find  the  weight  of  the  fluid  necessary  to  fill  a  mould 
or  vessel  into  which  the  block  exactly  fits  ;  or  we  may 
fill  a  vessel  with  the  fluid,  and  weigh  the  quantity 
which  runs  over  when  the  block  is  immersed  ;  or  we 
may  load  the  block l  until  it  neither  floats  nor  sinks  in 
the  fluid, —  the  weight  of  the  block  being  in  this  case 
equal  to  that  of  an  equal  bulk  of  the  fluid  (§  64). 
Other  methods  will  be  described  in  experiments  which 
follow.  The  density  of  a  fluid  is  always  calculated, 
like  that  of  a  solid,  by  dividing  its  weight  by  its  volume. 
We  have  seen  how  one  may  find  the  weight  of  a 
certain  quantity  of  a  fluid  equivalent  in  volume  to 
a  rectangular  block  ;  the  volume  of  the  fluid  in  ques- 
tion (being  equal  to  that  of  the  block)  is  calculated 
by  multiplying  together  the  length,  breadth,  and 

1  In  a  wooden  block,  auger-holes  bored  parallel  to  the  grain  may 
be  nearly  filled  with  lead,  and  closed  with  a  wooden  plug  even  with  the 
surface.  A  cube  measuring  10  cm.  each  way  and  weighing  998  q.  will 
be  found  useful  to  illustrate  the  density  of  water.  The  block  should 
be  coated  with  oil  or  other  material  impervious  to  water. 


f  2.]  WEIGHING  BY  TRIAL.  3 

thickness  of  the  block.  All  measurements  of  den- 
sity will  be  found  to  depend  more  or  less  directly 
upon  linear  dimensions  as  well  as  upon  weight. 

The  density  of  water  may  be  found,  approximately, 
by  any  of  the  methods  suggested  above  ;  but  the  ex- 
act measurement  of  the  density  of  water  is  one  of  the 
most  difficult  problems  in  physical  measurement.  "VVe 
shall  need  continually  to  refer  to  the  values  in  Table 
25,  which  have  been  obtained  by  combining  the  re- 
sults of  the  most  careful  observers.  The  student 
will  of  course  accept  these  values  in  preference  to 
any  which  he  himself  may  obtain ;  but  to  use  them 
intelligently,  he  must  thoroughly  understand  both 
what  they  represent  and  how  they  are  found.  He 
should  convince  himself  that  the  density  of  water  is 
not  far  from  unity ;  or  that,  in  other  words,  1  cu.  cm. 
of  water  weighs  nearly  1  g.  (see  §  6)  ;  and  he  should 
familiarize  himself  with  the  fundamental  method  of 
measuring  density  by  weight  and  linear  dimensions, 
applicable,  as  we  have  seen,  either  to  a  solid  or  to  a 
liquid.1  In  case  that  a  rectangular  block  is  used,  the 
necessary  data  are  its  weight  in 
grams,  and  its  length,  breadth, 
and  thickness  in  centimetres. 
The  observations  are  made  as 
stated  below. 

^[  2.  Determination  of  "Weight  FIG.  1. 

by  the  Method  of  Trial.  —  The  block  is  to  be  weighed 
with  rough  scales,  such  as  are  represented  in  Fig.  1, 
and  which  should  be  affected  by  a  decigram.  To 

1  See  the  Harvard  University  List  of  Chemical  Experiments,  Exp.  1. 


4  MEASUREMENT  OF  DENSITY.  [Exp.  1. 

select  the  weights  necessary  to  balance  a  given  body 
requires  in  general  many  trials.  The  number  of  trials 
may  be  greatly  reduced,  in  the  long  run,  by  a  strict 
adherence  to  the  method  here  described.  (See  §  35, 
2d  ed.)  We  first  place  the  block  on  one  scale-pan, 
and  a  single  weight,  which  we  judge  to  be  nearly 
equal  to  it,  on  the  other.  If  this  weight  is  too  small, 
that  is,  if  it  is  insufficient  to  lift  the  block,  we  add  to 
it  another  weight  of  about  equal  magnitude,  if  any 
such  exist  in  the  set  of  weights;  or  should  there 
be  no  weight  equal  to  the  first,  we  add  one  of  the 
next  greater  magnitude.  If  the  two  weights  to- 
gether fail  to  lift  the  block,  we  add  a  third  as  nearly 
equal  to  the  sum  of  the  other  two  as  may  be  conven- 
ient, and  thus  by  doubling  the  weight  in  one  scale-pan 
as  many  times  as  may  be  necessary,  we  find  a  quantity 
capable  of  lifting  the  load  in  the  other  scale-pan.  If 
on  the  other  hand,  the  first  weight  tried  lifts  the  block, 
that  is,  if  it  is  too  heavy,  we  substitute  for  it  one  half 
as  great,  if  any  such  be  contained  in  the  set ;  other- 
wise, the  largest  weight  less  than  half  of  the  first ; 
and  if  the  second  weight  is  too  great  we  substitute 
in  the  same  way  a  third  weight  not  greater  than  half 
of  the  second,  and  so  continue  to  halve  the  weight 
until  finally  it  is  lifted  by  the  block. 

The  weight  of  the  block  thus  becomes  known  be- 
tween two  limits.  We  next  try  a  weight  as  nearly 
half-way  between  these  limits  as  may  be  obtained  by 
the  addition  or  subtraction  of  one  weight  at  one  time, 
or  by  the  substitution  of  one  weight  for  another;  and 
thus  gradually  approximate  to  the  weight  of  the 


VERNIER  GAUGE. 


block  by  successively  halving  the  interval  between 
the  limits  known  to  contain  it. 

By  aimless  departures  from  this  method  of  approx- 
imation, the  number  of  trials  may  be  indefinitely  in- 
creased ;  but  certain  modifications  may  be  advisable 
when,  from  the  slow  motion  of  the  scales  or  from  any 
other  cause,  one  has  good  ground  to  think  that  the 
true  weight  has  been  nearly  found.  In  all  such  cases 
one  should  add  or  take  away  only  so  much  weight  as 
may  be  reasonably  expected  to  turn  the  scales. 

When  the  block  has  been  exactly  counterpoised  by 
weights,  it  should  be  transferred  to  the  other  scale- 
pan  and  balanced  against  the  same  weights  as  before. 
(See  §  44.)  If  the  scales  are  as  accurate  as  they  are 
"precise,"  (§  48,  2d  ed.)  the  equilibrium  will  not 
be  disturbed,  otherwise  a  readjust- 
ment of  the  weights  will  be  neces- 
sary. In  the  latter  case  the  average 
of  the  two  weighings  is  adopted  as 
the  true  weight  of  the  block.  (See 
Experiment  8.) 

*f[  3.  Determination  of  Length, 
Breadth,  and  Thickness  by  a  Vernier 
Gauge.  —  We  have  seen  in  ^[  2  how 
the  weight  of  a  block  can  be  found; 
it  remains  to  measure  its  length, 
breadth,  and  thickness,  in  order 
that  its  density  may  be  determined. 
A  Vernier  gauge  (Fig.  2)  is  suitable  for  this  pur- 
pose. To  obtain  great  accuracy  with  such  a  gauge, 
special  precautions  are  necessary  (see  Experiment 


FIG.  2. 


a 


MEASUREMENT  OF  DENSITY. 


[Exp.  1. 


19).  For  the  purposes  of  this  experiment,  however, 
it  will  be  sufficient  to  observe  that  the  distance  be- 
tween the  jaws  (c?  and  d)  is  directly  indicated  on 
the  main  scale  of  the  instrument  by  the  "  pointer  " 
or  "zero"  of  the  Vernier  scale  (6)  on  the  sliding 
piece  (a  b  <?),  to  which  the  jaw  (c)  is  attached.  To 
identify  the  zero  of  the  vernier,  we  bring  the  jaws 
(c>  and  <f)  into  contact ;  the  zero  of  the  vernier  should 
then  come  opposite  to  the  zero  of  the  main  scale. 
For  convenience  in  reading  the  vernier,  the  zero  is 
generally  placed  at  a  point  (6)  con- 
siderably beyond  the  movable  jaw 
(c)  ;  but  if,  as  in  the  figure,  the 
main  scale  begins  at  an  equal  dis- 
tance from  the  fixed  jaw  (d),  the 
readings  will  not  be  affected.  Evi- 
dently, in  such  a  gauge,  the  edge  of 
the  sliding  jaw  (c?)  cannot  be  used 
as  an  index. 

The  whole  number  of  millimetres 
between  the  jaws  is  equal  to  the 
number  of  the  first  millimetre  divis- 
ion below  the  zero  of  the  vernier, 
that  is,  between  it  and  the  zero  of 
the  main  scale.  The  tenths  of  mil- 
limetres above  this  whole  number 
may  be  read  from  the  vernier  as  explained  in  §  40. 

The  block  is  first  clamped  lengthwise  between  the 
jaws  of  the  gauge  as  in  figure  3,  and  ten  measure- 
ments are  thus  taken  at  different  points.  It  is  then 
clamped  so  as  to  obtain  in  a  similar  manner  ten 


FIG.  3. 


f  4.]  CORRECTIONS  NEGLECTED.  7 

measurements  of  its  breadth,  and  finally  ten  of  its 
thickness.  In  each  case  the  readings  are  made  to 
millimetres  and  tenths.  The  object  of  taking  a  large 
number  of  measurements  is  to  find  the  average  length, 
breadth,  and  thickness  with  a  degree  of  exactness 
(§  48)  corresponding  to  that  attained  in  the  weighing 
already  performed  (^[  2).  We  finally  calculate  the 
volume  and  density  of  the  block  as  explained  in  ^[  1. 

^[  4.  Corrections  Disregarded  in  Experiment  1.  — 
The  vernier  gauges  which  we  usually  employ  are 
supposed  to  read  correctly  at  0°  Centigrade ;  and 
hence  will  not  be  quite  accurate  at  ordinary  tempera- 
tures. For  instance,  if  the  gauge,  having  been  cooled 
by  melting  ice  to  0°,  is  fitted  to  the  block  as  in 
Fig.  3,  then  allowed  to  become  warm  through  con- 
tact with  the  air  of  the  room,  it  will  no  longer  fit 
the  block  as  closely  as  it  did,  owing  to  expansion  of 
the  metal  by  heat.  The  block,  though  really  un- 
changed in  size,  will  appear  to  be  somewhat  smaller 
than  before.  This  effect  of  expansion  is  barely  per- 
ceptible ;  but  we  tend,  nevertheless,  to  underestimate 
all  the  dimensions  of  the  block,  and  hence  also  its 
volume.  With  brass  gauges  at  20°,  the  error  in  the 
volume  would  amount  to  about  1  part  in  900  (see 
Table  8  6,  also  §  83). 

Another  source  of  error  lies  in  the  fact  that  the 
weighings  ai'e  made  in  air,  and  not  in  vacua  (§  65). 
In  the  case  of  a  body  weighing  about  one  gram  to  the 
cubic  centimetre,  it  is  found  (see  Table  21),  that  the 
atmosphere  exerts  a  buoyant  action  which  apparently 
deprives  it  of  about  one  900th  of  its  weight.  We 


8  NICHOLSON'S  HYDROMETER.  [Exp.  2. 

should  therefore  underestimate  both  the  weight  and 
the  volume,  in  such  a  case,  in  the  same  proportion; 
and  the  density  obtained  by  dividing  the  one  by  the 
other  would  not  be  affected.  Even  when  the  correc- 
tions in  this  experiment  do  not,  as  above,  completely 
offset  one  another,  they  generally  amount  to  less  than 
one  part  in  a  thousand,  and  may  be  neglected  in  com- 
parison with  errors  of  observation.  (See  §  24.) 

EXPERIMENT  II. 

TESTING   A    HYDROMETER. 

^[  5.    Determination    of  the    Sensitiveness    of    a   Hy- 
drometer.—  A  Nicholson's  hydrom- 
eter is  to  be  loaded  as  in  Fig.  4,  by 
placing  weights  in  the  upper  pan, 
a,  until   a   small  ring  round   the 
lower  part  of  the  wire  stem  sinks 
just  beneath   the   surface   of  the 
water ;    then    small   weights    are 
added,  say  5  centigrams,  until  by 
the    sinking    of 
the  instrument, 
another   ring 
round  the  upper 
part  of  the  stem 
is  brought  just 
below  the  water 
level.     The  dis-  Fi(>.  4. 

tance  between  the  two  rings,  through  which  the  hy- 
drometer sinks  under  the  action  of  the  weight  added, 


IT  5.]  PKECAUTIONS.  9 

is  estimated  roughly  by  a  small  millimetre  scale.  We 
now  calculate  the  effect  of  one  centigram  in  sinking 
the  instrument.  This  is  called  the  sensitiveness 
(§  41)  of  the  hydrometer,  and  is  useful  in  determin- 
ing the  degree  of  precision  with  which  the  adjust- 
ments of  the  instrument  should  be  made  (see  §  48). 
Thus  if  the  effect  of  one  centigram  is  distinctly  per- 
ceptible, we  should  try  to  avoid  errors  even  less  than 
a  centigram  in  magnitude. 

In  using  a  Nicholson's  hydrometer,  several  precau- 
tions should  be  observed.  It  frequently  happens 
that  through  friction  against  the  sides  of  the  vessel, 
or  through  capillary  phenomena  where  the  surface  of 
the  water  meets  the  stem,  the  hydrometer  is  unaf- 
fected by  any  slight  change  iu  the  load.  To  avoid 
the  first  difficulty,  the  instrument  should  be  kept 
floating  in  the  middle  of  the  jar,  by  the  use  of  a 
guide  of  some  sort.  Such  a  guide  may  be  con- 
veniently constructed  of  wire,  as 
in  Fig.  5.  To  avoid  the  uncer- 
tainty of  capillary  action,  the  stem 
of  the  hydrometer  should  be  kept 
wet,  by  a  camel's-hair  brush,  for 
at  least  a  centimetre  above  the 
water  level.  FlG-  5- 

In  water  freshly  drawn  bubbles  of  air  are  apt  to 
form,  clinging  to  the  sides  of  the  hydrometer.  These 
should  be  removed  by  the  same  brush.  The  forma- 
tion of  air  bubbles  may  generally  be  prevented  by 
using  either  distilled  water,  or  water  which  has  been 
standing  for  some  time  in  the  room. 


K)  NICHOLSON'S  HYDROMETER.  [Exp.  2. 

It  is  important  to  keep  the  upper  part  of  the  stem, 
the  pan,  and  the  weights  absolutely  dry.  The  guide 
(Fig.  4)  should  prevent  the  hydrometer  from  sinking 
completely  below  the  surface.1 

^f  6.  Accurate  Adjustment  of  a  Nicholson's  Hydrom- 
eter. __  A  mark  is  made  near  the  middle  of  the  stem 
of  the  hydrometer  and  the  load  is  altered,  a  centi- 
gram at  a  time,  until  this  mark  is  floated  as  nearly 
as  possible  in  line  with  the  surface  of  the  water.  If 
a  glass  jar  is  used,  it  is 
better  to  sight  this  mark 
by  the  under  surface  of 
the  water,  as  shown  in 
Fig.  6. 

In  the  absence  of 
weights  smaller  than  one 
centigram,  we  estimate 

and  record  fractions  of  a  centigram  as  follows : 
when  the  mark  is  floated  exactly  on  a  level  with 
the  surface  of  the  water,  the  fact  is  expressed  by 
placing  a  cipher  in  the  third  decimal  place  (be- 
longing to  the  milligrams).  If,  however,  a  given 
weight  fails  to  sink  the  mark  to  this  level,  while 
the  addition  of  one  centigram  sends  it  as  much 
below  the  surface  as  it  was  before  above  it,  half  a 


1  A  sheet  of  cardboard  or  metal  with  a  hole  in  the  middle,  is  rec- 
ommended by  some  authorities  (see  Pickering's  Physical  Manipu- 
lation, Article  45)  to  serve  as  a  guide,  and  at  the  same  time  to 
prevent  the  weights  from  falling  into  the  water.  A  student  relying 
upon  this  safeguard  is  apt,  however,  not  to  acquire  a  sufficient  degree 
of  skill  to  prepare  him  for  the  manipulations  of  a  delicate  balance. 
(Exps.  6-14.) 


1  7.]  EFFECTS  OF  TEMPERATURE.  11 

centigram  or  5  milligrams  is  obviously  the  weight 
to  be  added  ;  hence  the  original  weight  should  be 
followed  by  a  5  instead  of  a  0  in  the  last  place. 
Thus  if  with  25.99  g.  the  mark  is  2  mm.  above  the 
surface  of  the  water,  and  with  26.00  g.  it  is  2  mm. 
below  it,  the  weight  sought  must  be  25.995  g. 
Again,  if  the  lesser  of  two  weights  differing  by 
one  centigram  is  evidently  nearer  than  the  other 
to  the  weight  desired,  we  substitute  a  figure  2  or 
a  3  for  the  5  in  the  last  place,  or  if  the  greater 
weight  is  more  accurate,  we  write  a  7  or  an  8  in- 
stead. Any  distinct  information  of  this  kind  should 
always  be  recorded  when  possible,  by  means  of  a 
figure  in  the  last  place,  even  if  that  figure  be  ex- 
tremely doubtful  (§  55).  Closer  estimates  will 
hardly  be  justified  in  the  case  of  a  Nicholson's 
hydrometer. 

^[  7.  Effect  of  Temperature  on  a  Nicholson's  Hydrom- 
eter. —  The  temperature  of  the  water  in  the  jar  is 
now  taken.  The  water  is  then  cooled  with  ice  to 
about  10°,  and  the  weight  required  to  balance  the 
hydrometer  is  determined  as  before,  with  a  new 
observation  of  temperature.  Then  the  jar  is  filled 
with  tepid  water  (at  about  30°)  and  the  experiment 
is  repeated.  A  comparison  of  the  different  results 
shows  how  much  the  buoyancy  of  water  is  affected 
by  temperature.  For  this  purpose  the  observations 
which  we  have  now  obtained  at  three  different  tem- 
peratures are  to  be  represented  graphically  on  co- 
ordinate paper  by  three  points,  A  B  and  (7,  as 
explained  in  §  59,  and  through  these  points  the 


12  NICHOLSON'S   HYDROMETER.  [Exp.  2. 

curve  A  B  C  is  to   be  drawn  with  a  bent  ruler. 
(See  Fig.  7.) 

The  ambitious  student  may  supplement  this  ex- 
periment by  using  water  hotter  than  30°  and  colder 
than  10°,  also  water  at  intermediate  temperatures. 
He  will  thus  obtain  data  for  plotting  a  more  com- 


FIG.  7. 


plete  curve  than  that  shown  in  the  figure.  This 
curve,  if  we  neglect  the  expansion  of  the  metal  of 
which  the  hydrometer  is  composed,  represents  the 
relative  buoyancy,  or  (see  §  64)  the  relative  density, 
of  water  at  different  temperatures. 


119.] 


WEIGHING  WITH  A  HYDROMETER. 


13 


a 


EXPERIMENT  III. 

WEIGHING   WITH   A    HYDROMETER. 

^|  8.  Determination  of  Weight  in  Air  by  a  Nicholson's 
Hydrometer.  —  From  the  results  of  Experiment  2  it 
is  possible  to  find  (see  §  59)  the  weight  necessary 
to  sink  a  hydrometer  to  a  given 
mark  in  water  of  any  ordinary 
temperature.  It  is  obvious  that 
in  all  determinations  with  a 
Nicholson's  hydrometer,  the  tem- 
perature of  the  water  must  be 
observed  at  the  time  of  weighing. 
To  find  the  weight  of  a  body, 
place  it  in  the  upper  pan  (a,  Fig. 
8),  and  with  it  enough  weights 
from  the  box  to  sink  it  to  the 
same  mark  as  before.  Evidently 
less  weight  will  be  required 
than  at  the  same  temperature 
without  the  body,  and  the  dif- 
ference will  be  equal  to  the 
weight  of  the  body  in  question. 

^[  9.  Reasons  for  Neglecting  Corrections  for  the 
Buoyancy  of  Air.  —  Since  the  air  buoys  up  both  the 
brass  weights  and  the  body  used,  the  result  of  this 
experiment  is  what  we  call  the  apparent  weight  of 
the  body  in  air  (§  65).  The  amount  of  this  buoy- 
ancy depends  (see  §  68)  in  one  case  upon  the  density 
of  the  brass  weights,  in  the  other  case,  upon  that  of 


FIG. 


14  NICHOLSON'S   HYDROMETER.  [Exp.  4. 

the  body  in  question  ;  hence  if  these  two  densities 
are  approximately  equal,  the  air  will  exert  nearly  the 
same  force  in  both  cases.  The  result,  obtained  as 
we  have  seen  by  difference,  will  not  therefore  be 
affected  to  an  appreciable  extent. 

For  the  purposes  of  this  and  other  experiments 
which  follow,  we  choose  ten  steel  balls,  perfectly 
round  and  uniform  in  size,  such  as  are  used  in  the 
bearings  of  the  front  wheel  of  a  bicycle.  The  density 
of  these  balls  (7 -8)  is  not  far  from  that  of  the  brass 
weights  (84),  and  it  will  be  seen  by  reference  to 
Table  21  that  the  correction  for  the  buoyancy  of 
air  may  be  wholly  disregarded. 


EXPERIMENT  IV. 

WEIGHING   IN  WATER   WITH    A    HYDROMETER. 

^j  10.  Determination  of  Specific  Gravity  by  a  Nich- 
olson's Hydrometer.  —  The  steel  balls  used  in  the  last 
experiment  are  now  to  be  placed  in  the  lower  pan  of 
the  hydrometer  (L  Fig.  9),  which  is  lifted  by  the 
stem  out  of  the  water  for  this  purpose.  The  instru- 
ment is  then  balanced  with  weights,  and  the  tem- 
perature of  the  water  observed  as  in  the  last  two 
experiments. 

In  lowering  the  hydrometer  into  the  jar,  care  must 
be  taken  to  remove  with  a  camel's-hair  brush  all 
bubbles  of  air  from  the  steel  balls,  as  well  as  from  the 
sides  of  the  hydrometer,  and  also,  of  course,  not  to 
spill  any  of  the  balls.  In  the  adjustment  of  weights 


SPECIFIC  GRAVITY. 


lo 


the  same  precautions  must  be  used  as  in  the  last  two 
experiments.     We  have  already  obtained  the  weight 
of  the  steel  balls  in  air  (^[  8)  ;  we  find  similarly  their 
weight  in  water  from  the  results 
of  Experiments   2   and   4,  and 
finally   their    apparent    specific 
gravity  (see  §  66). 

T[  11.  Use  of  the  Methods  of 
Substitution  and  Multiplication. 
—  It  will  be  noted  that  in  Ex- 
periment 3  the  unknown  weight 
of  a  body  takes  the  place  of  a 
known  weight  of  brass  used  in 
Experiment  2  ;  the  one  is  in  fact 
substituted  for  the  other.  The 
method  of  finding  the  weight  of 
a  body  by  a  Nicholson's  hy- 
drometer is  therefore  essentially 
a  method  of  substitution  (§  43).  This  statement 
also  applies  to  the  determination  of  weight  in  water 
by  the  same  instrument ;  for  the  weight  of  a  body  in 
water  is  here  substituted  for  a  known  weight  of  brass 
in  air.  The  errors  committed  with  a  Nicholson's  hy- 
drometer depend  upon  the  peculiarities  of  the  instru- 
ment itself,  rather  than  upon  the  quantities  weighed. 
We  are  in  fact  liable  to  the  same  error  in  weighing 
one  bicycle  ball  as  in  weighing  ten.  The  proportion 
which  the  error  bears  to  the  total  quantity  weighed  is, 
however,  diminished  when  this  quantity  is  increased. 
The  use  of  a  large  number  of  bicycle  balls  for  the 
determination  of  specific  gravity  in  Experiments  3 


FIG.  (J. 


1G  NICHOLSON'S  HYDROMETER.  [Exp.  4. 

and  4  is  a  good  example  of  the  accuracy  gained  by 
the  method  of  multiplication  (§  39). 

^[12.  Corrections  Disregarded  in  Experiment  4.  — 
In  the  last  experiment  we  disregarded  the  effects  of 
the  buoyancy  of  air  on  the  steel  balls  and  on  the 
brass  weights,  because  these  effects  were  so  nearly 
equal,  both  being  in  air.  Here,  however,  the  balls 
are  in  water  and  the  weights  in  air. 

There  is,  therefore,  nothing  to  compensate  for  the 
buoyancy  of  air  on  the  brass  weights.  It  is  seen  by 
reference  to  §  65  that  7  grams  of  brass  are  buoyed 
up  by  the  air  with  a  force  of  about  1  milligram ;  and 
as  a  Nicholson's  hydrometer  can  float  only  about  4 
times  7,  or  28  grams,  the  effect  of  buoyancy  on  the 
weights  cannot  be  greater  than  4  milligrams.  This 
error  may  generally  be  disregarded  in  comparison 
with  the  errors  of  observation.  The  manner  of 
applying  a  correction  for  the  buoyancy  of  air  is  ex- 
plained in  Experiments  8  and  9,  also  in  §§  65-68. 

In  calculating  apparent  specific  gravity,  no  correc- 
tions need  be  taken  into  account;  but  the  result 
should  be  expressed  as  the  apparent  specific  gravity 
of  a  given  body  at  a  given  temperature  referred  to 
water  at  a  given  temperature.  The  result  will  be 
affected  somewhat  by  the  density  of  the  air,  but 
hardly  to  a  perceptible  extent.  The  student  is  ad- 
vised, as  a  matter  of  habit  simply,  to  note  the  condi- 
tions of  the  atmosphere  in  which  his  weighings  are 
performed  (see  Experiment  5). 


IT  13.] 


THE  BAROMETER. 


17 


:f 


EXPERIMENT  V. 

ATMOSPHERIC   DENSITY. 

^[13.  Determination  of  Barometric  Pressure.  —  The 
three  conditions  of  the  atmosphere  which  affect  the 
results  of  physical  measurement  are  barometric  press- 
ure, temperature,  and  humidity.  Let  us  first  con- 
sider how  barometric  pressure  is  observed.  A  very 
rough  but  serviceable  form  of  mercurial  barometer 
consists  simply  of  a  glass-tube  (a  5,  Fig.  10),  which, 
having  been  filled  with  mercury,1 
is  inverted  in  a  cistern  of  mercury  Q 
(6).  The  mercury  sinks  in  the 
closed  end  of  the  tube  to  a  level 
a,  above  which  there  will  be  a 
nearly  perfect  vacuum.2  As  there 
is  no  pressure  at  a,  to  counter- 
act the  atmospheric  pressure  be- 
low, the  mercury  stands  in  the 
tube  at  a  level  (a)  above  the  level 
(i)  in  the  cistern.  It  is  found  by 
experiment3  that  the  atmospheric  /.  1 
pressure  is  transmitted  through 
the  cistern  of  mercury  and  the 
open  end  of  the  tube  to  a  point,  5,  on  a  level  with 
the  surface  of  the  mercury  in  the  cistern.  The 
atmospheric  pressure  is  accordingly  determined  by 

1  The  tube  and  the  mercury  must  be  perfectly  clean  and  dry. 
For  cleaning  mercury,  see  Pickering's  Physical  Manipulation,  I.  9. 

2  The  "  Torricellian  vacuum."  «  See  §  62. 


FIG.  10. 


18  ATMOSPHERIC   DENSITY.  [Exp.  5. 

the  length,  a  6,  of  the  column  of  mercury  which  it 
sustains.1  The  distance  a  b  can  be  measured  by 
means  of  a  graduated  wooden  rod,  by  which  the  tube 
is  supported  in  a  vertical  position.  The  level  b  is 
first  sighted  in  the  ordinary  manner  with  care  to 
avoid  parallax  (§  25)  ;  and  the  reading  thus  found 
is  subtracted  from  that  of  the  level  a,  obtained  in  a 
similar  manner  (see  §  32). 

In  the  case  of  a  standard  mercurial  barometer  the 
lower  end  of  the  column  of  mercury  should  always 
be  looked  at  first,  else  a  considerable  error  is  likely 
to  arise  (§  32)  ;  for  even  when  the  barometer  ends  in 
a  large  cistern  of  mercury,  the  level  in  this  cistern 
must  vary  somewhat  as  more  or  less  mercury  rises 
into  the  tube.  In  some  barometers  this  rise  and  fall 
is  compensated  by  turning  a  screw  (d,  Fig.  11). 
This  raises  or  lowers  the  mercury  in 
the  cistern,  and  when  a  certain  steel 
or  ivory  point  (a,  Fig.  11),  just  touches 
its  own  reflection,  the  level  of  the  mer- 
cury is  known  to  be  at  the  right  height. 
When  the  lower  end  of  the  mercurial 
column  in  the  tube  has  been  thus  ad- 
justed, the  height  of  the  upper  end  is 
usually  read  by  a  movable  sight,  pro- 
vided with  a  vernier  (§  40).  The  lower 
edge  of  the  sight  is  to  be  set  on  a  level 
with  the  highest  part  of  the  mercurial  column,  so  as 
to  appear  to  be  tangent  to  the  meniscus  or  curved 
surface  of  the  mercury  (Fig.  12,  a).  To  avoid  par- 
1  See  §  63. 


1  14.]  CORRECTIONS  OF  A  BAROMETER.  19 

allax  (§  25),  a  double  sight  is  frequently  used,  con- 
sisting of  two  edges  in  the  same  horizontal  plane, 
one  in  front  of,  the  other  behind  the  mercurial 
column.  The  student  should  find  by  direct 
measurement  whether  the  distance  from  the 
zero-point  (a,  Fig.  11),  to  the  lower  edge  of 
the  sight  (a,  Fig.  12)  is  indicated  correctly 
upon  the  scale  of  the  barometer.  If  the 
reading  of  the  barometer  is  in  inches,  it 
may  be  reduced  to  centimetres  conveniently 
by  Table  16.  FlG-  12- 

Aneroid  barometers  are  generally  constructed  so  as 
to  agree  very  closely  with  mercurial  barometers. 
They  will  be  found  accurate  enough  for  correcting 
the  results  of  most  physical  measurements.  If  an 
Aneroid  barometer  is  to  be  used,  the  student  should 
compare  its  indication  with  that  of  a  mercurial  baro- 
meter, determined  as  explained  above. 

^[  14.  Corrections  of  a  Barometer.  —  A  small  quan- 
tity of  air  almost  always  finds  its  way  sooner  or  later 
into  the  space  above  the  mercury  in  a  barometer  (a, 
Fig.  10),  where  it  causes  a  slight  depression  of  the 
column.  To  test  a  barometer  for  air,  we  tilt  the  tube 
ab  (Fig.  10)  into  a  new  position  af  5,  being  careful 
to  keep  the  mercury  in  the  cistern  at  a  constant  level, 
b,  either  by  raising  the  cistern  or  by  adding  more 
mercury  to  compensate  for  that  which  flows  into  the 
tube.  In  the  absence  of  air>  the  mercury  should 
follow  the  horizontal  line  a  a',  and  should  completely 
fill  the  tube  when  the  inclination  is  sufficiently  in- 
creased. 


20  ATMOSPHERIC   DENSITY.  [Exp.  5. 

A  simple  way  of  correcting  for  air  in  a  barometer 
is  to  adjust  the  angle  a'  b  a  (Fig.  10)  by  trial,  so  that 
the  space  above  a'  is  half  that  above  a.  By  thus 
reducing  the  air  to  half  its  original  volume,  the 
pressure  will  be  doubled;1  hence  a!  will  be  as  much 
below  a  as  a  is  below  its  proper  level.  By  measuring 
the  difference  between  the  levels,  a  and  a',  we  find 
accordingly  the  correction  for  air.  A  correction  of  2 
or  3  mm.  may  be  disregarded,  as  it  will  probably  be 
offset  by  other  corrections  which  the  accuracy  of  the 
instrument  will  not  justify  us  in  considering.  In 
case  the  correction  is  much  larger  than  this,  the  ba- 
rometer should  be  refilled  with  mercury.  The  filling 
of  a  standard  barometer  should  be  attempted  only  by 
a  skilled  workman.  Unless  perfectly  free  from  air, 
such  a  barometer  is  little  better  than  the  rough  instru- 
ment shown  in  Fig.  10. 

In  all  exact  readings  of  a  barometer,  the  three 
following  corrections  are  usually  applied :  (a)  for  ex- 
pansion, (b)  for  capillary  depression,  and  (c)  for  the 
pressure  of  mercurial  vapor.2  The  temperature  of 
the  mercury  in  a  barometer  is  found  by  a  thermometer 
beside  it.  Let  t  be  this  temperature,  reduced  if 
necessary  to  the  Centigrade  scale  (see  Table  39), 
and  let  h  be  the  height  in  centimetres  of  the  mer- 
curial column  ;  then  the  correction  for  expansion  is 
.00018  ht,  which  is  to  be  subtracted  from  the  ob- 
served height.  The  object  of  this  correction  is  to  find 

This  follows  from  the  law  of  Boyle  and  Mariotte  (§  79). 
2  The  reduction  of  a  barometric  reading  "to  the  sea  level  "  is  not 
required  for  the  purposes  of  physical  measurement. 


1  14.]  COEKECTIONS  OF  A  BAROMETER.  21 

how  high  the  mercury  would  stand  if  its  temperature 
were  0°  Centigrade.  Since  1  cm.  of  mercury  when 
heated  1°  Centigrade  expands  by  the  amount  .00018 
cm.  (see  Table  11),  h  cm.  would  expand  h  times  as 
much ;  and  h  cm.  heated  t°  would  expand  Jit  times  as 
much,  whence  we  obtain  the  correction  in  question. 
At  the  ordinary  temperature  of  a  room  (20°),  and  at 
the  barometric  pressure,  75  cm.,  this  correction  for 
expansion  would  be  .00018  X  20  X  75  cm.  =  2.7  mm. 
It  is  therefore  useless  to  read  a  barometer  (as  is  often 
done)  to  tenths  or  hundredths  of  a  millimetre,  when 
no  correction  for  temperature  is  made.  The  correc- 
tion given  above  may  be  applied  to  barometers  with 
wooden  or  glass  scales,  the  expansion  of  which  may 
be  neglected.  When,  however,  the  body  of  the  in- 
strument consists  of  steel,  the  coefficient  .00017 
should  be  used  instead  of  .00018  ;  and  if  the  barome- 
ter is  mounted  in  brass  or  white  metal,  the  factor 
.00016  will  be  still  more  accurate.  These  numbers 
represent  the  difference  of  expansion  between  the 
mercury  and  the  scale  by  which  it  is  measured.  For 
more  accurate  values  see  Table  18  a. 

When  the  tube  of  a  barometer  is  less  than  a  centi- 
metre in  diameter,  there  is  found  to  be  a  perceptible 
depression  of  the  mercurial  column  due  to  "  capillar- 
ity," or  "surface  tension,"  the  general  nature  of 
which  will  be  investigated  farther  in  Experiment  67. 
The  internal  diameter  of  the  tube  should  be  found  if 
possible  by  measuring  a  plug  which  fits  it  in  the  part 
where  the  column  of  mercury  ends  (see  a,  Fig.  10). 
A  different  method  of  calibration  will  be  considered 


22  ATMOSPHERIC  DENSITY.  [Exp.  5. 

in  Experiment  26.  Wheii  the  internal  diameter  is 
known,  the  correction  for  capillarity  may  be  found 
roughly  from  Table  18  b.  Thus  for  a  tube  5  mm.  hi 
diameter,  in  which  the  height  of  the  mercury  menis- 
cus is  unknown,  the  capillary  depression  may  be  taken 
as  1.5  mm.  In  various  barometers  which  are  con- 
structed so  that  the  internal  diameter  cannot  be 
measured,  we  generally  assume  that  the  instrument- 
maker  has  allowed  for  capillarity  in  adjusting  his 
scale,  and  we  therefore  neglect  this  correction.  It  is 
customary,  also,  to  neglect  the  effect  of  capillary 
phenomena  in  the  cistern  of  mercury. 

Owing  to  the  evaporation  of  mercury  into  the 
space  above  it  in  the  tube  of  the  barometer,  that 
space  is  never  quite  empty.  The  quantity  of  mer- 
curial vapor  which  it  contains  is  found  to  increase 
when  the  temperature  increases,  and  also  the  press- 
ure which  it  exerts.  To  allow  for  the  slight  de- 
pression of  the  mercurial  column  due  to  this  cause, 
Table  18  c  has  been  constructed  from  the  results  of 
actual  observation.  Thus  for  a  temperature  of  20°, 
we  find  that  the  mercurial  column  is  depressed  to  the 
extent  of  0.02  mm.  by  the  pressure  of  its  own  vapor. 

We  have  found  in  a  particular  case  that  2.7  mm. 
should  be  subtracted  from  the  observed  height  of  a 
barometer  on  account  of  expansion  ;  that  1.5  mm. 
should  be  added  for  capillarity  and  also  0.02  mm.  to 
offset  the  pressure  of  mercurial  vapor.  The  resulting 
correction  is  1.18  mm.,  to  be  subtracted  ;  or  let  us 
say,  —1.2  mm.  nearly.  The  student  who  employs  a 
mercurial  barometer  should  find  in  the  same  way  an 


If  15.]  TEMPERATURE  AND  HUMIDITY.  23 

average  correction  for  it.  If  an  Aneroid  is  used, 
such  a  correction  is  found  by  comparing  one  reading 
at  least  with  the  corrected  reading  of  a  mercurial  ba- 
rometer. In  the  course  of  experiments  which  follow, 
readings  of  the  barometer  are  needed  only  for  slight 
corrections  in  the  results  of  physical  measurement. 
By  applying  to  the  barometer  an  average  correction, 
much  labor  will  be  saved,  and  the  error  introduced 
will  be  insignificant. 

^|  15.  Determination  of  Atmospheric  Temperature  and 
Humidity.  —  The  temperature  of  the  air  of  a  room 
may  be  determined,  with  a  sufficient  degree  of  ac- 
curacy for  most  purposes,  by  an  ordinary  mercurial 
thermometer,  the  reading  of  which  may  be  reduced 
from  the  Fahrenheit  to  the  Centigrade  scale  by  Table 
39.  The  thermometer  should  be  brought  as  near  as 
may  be  practicable  to  the  place  where  the  tempera- 
ture is  required.  It  should,  for  instance,  be  inside  of 
the  balance  case  in  very  delicate  weighings.  It  must 
not,  however,  be  exposed  to  the  rays  of  the  sun,  nor 
for  any  length  of  time  to  the  heat  radiated  by  a  lamp 
or  by  the  human  body.  When  the  greatest  accuracy 
is  desired,  the  bulb  of  the  thermometer  should  be 
protected  from  radiation  to  or  from  surrounding 
objects,  by  a  shield  of  polished  metal. 

The  humidity  of  the  atmosphere  is  most  conven- 
iently determined  by  a  class  of  instruments  of  which 
the  hygrodeik  is  an  example.  The  indications  of 
these  instruments  depend  upon  the  cooling  produced 
by  evaporation  (see  §  88).  It  is  found  that  when 
the  bulb  of  a  thermometer  is  covered  with  wet  wicking 


24 


ATMOSPHERIC  DENSITY. 


[Exp.  5. 


FIG.  13. 


(a,  Fig.  13),  its  reading  differs  from  that  of  an  ordi- 
nary thermometer  (6)  by  an  amount  depending  upon 
the  dryness  of  the  air.  When  the  air  is  completely 
saturated  with  moisture,  as  in  a 
dense  fog,  there  is  no  evaporation 
from  the  wet  bulb,  hence  the  two 
thermometers  agree;  if  the  air, 
however  is  heated,  the  fog  disap- 
pears, evaporation  begins,  and 
the  wet-bulb  does  not  rise  so 
high  as  the  diy-bulb  thermome- 
ter. On  the  other  hand,  when 
the  air  of  the  room  is  cooled 
sufficiently,  either  fog  is  formed 
or  dew  is  precipitated  on  various  objects  ;  and  the 
two  thermometers  again  agree.  The  temperature  at 
which  this  occurs  is  called  the  dew-point,  and  is  cal- 
culated from  the  readings  of  the  wet  and  dry-bulb 
thermometers  by  reference  to  Table  15,  or  by  a  spe- 
cial mechanical  device,  for  the  operation  of  which 
directions  are  usually  furnished  by  the  instrument- 
maker. 

^[16.  Observation  of  the  Dew-point.  —  Unless  a  hy- 
grodeik  is  known  to  give  accurate  indications,  the 
latter  should  be  confirmed  by  a  direct  determination 
of  the  dew-point,  as  follows :  a  polished  metallic 
vessel  is  partly  filled  with  water,  and  as  much  ice 
and  salt  are  added  as  may  be  necessary  to  make  a 
film  of  moisture  condense  on  the  surface.  The  tem- 
perature at  which  this  first  occurs  is  just  below  the 
dew-point.  Soon,  however,  the  contents  of  the  ves- 


T  17.]  DEW-POINT.  25 

sel  become  warmer  through  contact  with  the  air,  and 
the  film  begins  to  disappear.  The  temperature  is 
now  a  little  above  the  dew-point.  By  observing 
carefully  a  thermometer  with  which  the  cold  con- 
tents of  the  vessel  are  continually  stirred,  the  dew- 
point  may  be  determined  within  two  limits,  differing 
by  less  than  one  degree. 

Care  must  be  taken  not  to  breathe  on  the  metallic 
vessel,  since  the  breath  is  much  damper  than  the  air 
of  the  room  ;  and  as  there  is  more  or  less  evaporation 
from  all  parts  of  the  human  body,  even  the  hand 
should  be  kept  as  far  away  as  possible. 

^[  17.  Relation  of  Relative  Humidity  to  Dew-point. 
The  actual  amount  of  moisture  in  a  given  quantity 
of  air  has  been  determined  by  extracting  it  through 
the  action  of  certain  hygroscopic  substances,  such  as 
chloride  of  calcium,  and  measuring  the  gain  in  their 
weight.  It  is  found  that  hot  air  can  hold  more 
moisture  without  forming  fog  than  cold  air.  We 
have  a  common  instance  in  the  air  of  a  room  which, 
though  apparently  dry  while  warm,  deposits  moisture 
upon  the  window-panes  by  which  it  is  cooled.1  The 
ratio  of  the  amount  of  moisture  actually  held  in  the 
air  (at  a  given  temperature)  to  the  maximum  amount 
(which  can  be  held  at  that  temperature)  is  called 
the  relative  humidity  of  the  air.  The  relations  be- 
tween temperature,  dew-point,  and  relative  humidity 
do  not  follow  any  simple  law ;  but  if  any  two  of 
these  quantities  are  given,  the  third  may  be  found  by 

1  For  a  further  illustration  see  list  of  Experiments  in  Elementary 
Physics,  published  by  Harvard  University,  Exercise  22. 


ATMOSPHERIC  DENSITY. 


[Exp.  5. 


referring  to  Table  15,  containing  the  results  of  various 
experiments. 

It  may  be  noted  that  the  dew-point  depends  solely 
upon  the  amount  of  moisture  in  the  air;  that  dry 
air  has  a  lower  dew-point  and  less  relative  humidity 
than  moist  air  at  the  same  temperature,  while  for  a 
given  dew-point  the  relative  humidity  increases  with 
a  fall  of  temperature,  until  fog  is  finally  formed,  or 
decreases  as  it  becomes  warmer  until  the  air  is  prac- 
tically dry.  It  should  also  be  noted  that  dry  air  is 
denser  than  moist  air.  We  must  regard  the  latter  as 
a  mixture  of  air,  not  with  water,  but  with  steam, 
which  is  only  about  two-thirds  as  heavy  as  air. 
Hence  in  Table  20  the  correction  for  moisture  is 
negative. 

^|  18.  Determination  of  Atmospheric  Density  by 
means  of  a  Barodeik.  —  From  the  temperature,  pres- 
sure, and  humidity  of 
the  atmosphere,  the  de- 
termination of  which 
has  been  explained 
above,  the  density  of 
air  may  be  calculated 
by  the  data  of  Tables 
19  and  20.  Whenever 
great  accuracy  is  de- 
sired this  calculation 
FIG.  14. 

must     be     performed. 

For  most  purposes,  however,  the  density  of  the  at- 
mosphere may  be  found  from  a  single  observation  of 
a  barodeik  (Fig.  14),  the  principle  of  which  is  spoken 


1f  19.]  MANIPULATION   OF  A  BALANCE.  27 

of  in  §  71.  It  is  important  to  compare  the  indication 
of  the  instrument  in  at  least  one  case  with  the  calcu- 
lated density  of  the  atmosphere.  A  reading  of  the 
barodeik  should  accompany  every  weighing  in  which 
more  than  three  figures  are  to  be  preserved,  except 
when  the  pressure,  temperature,  and  dew-point  have 
been  determined. 


EXPERIMENT  VI. 

TESTING   A    BALANCE. 

TJ  19.  Manipulation  of  a  Balance.  —  The  delicacy 
of  a  balance  depends  upon  the  sharpness  of  the 
knife-edges  (a  and  c,  Fig.  15)  from  which  the  pans 
are  suspended,  also  upon  the  sharpness  of  the  central 
knife-edge  (6)  upon  which  the  beam  (a  c)  turns.  In 
order  that  these  edges  may  not  become  dull,  the  pans 
should  be  supported  by  some  mechanical  device  at  all 
times  except  when  an  observation  is  actually  being 
taken.  It  is  particularly  important  that  they  should 
be  so  supported  when  they  are  being  loaded  or  un- 
loaded, or  when  the  balance  is  liable  to  be  jarred  in 
any  other  manner.  In  an  ordinary  prescription  bal- 
ance (Fig.  15),  the  pans  rest  upon  the  bottom  of  the 
case  when  the  instrument  is  not  in  use.  Such  a 
balance  is  thrown  into  operation  by  turning  a  milled 
head  outside  of  the  case.  The  beam  is  thus  raised  as 
slowly  as  possible,  so  as  not  to  injure  the  knife-edges 
by  suddenly  throwing  weight  upon  them.  It  is  not 
necessary  in  every  case  to  raise  the  beam  as  far  as  it 


28 


THE  BALANCE.  [Exp.  6. 


will  go.  As  soon  as  the  pointer  moves  decidedly  to 
one  side  or  the  other,  the  beam  should  be  slowly 
lowered  again.  In  other  cases  a  prolonged  observa- 
tion of  the  pointer  must  be  made  in  order  to  decide 
in  which  direction  the  beam  tends  to  incline.  During 
such  observations  the  beam  should  be  raised  to  its 
fullest  extent.  Whenever  accuracy  is  desired,  the 


FIG.  15. 

door  of  the  balance  case  should  be  closed,  in  order  to 
cut  off  currents  of  air  ;  in  fact,  the  door  should  never 
be  opened  except  when  the  purposes  of  manipulation 
actually  require  it.  This  precaution  is  necessary  to 
protect  the  instrument  from  moisture  and  dust,  and 
is  especially  important  when  the  air  within  the  bal- 
ance case  is  kept  artificially  dry  by  chloride  of  cal- 


f  19.]  MANIPULATION  OF   A  BALANCE.  29 

ciurn  or  other  hygroscopic  material.  The  glass  case 
should  be  cleaned  when  necessary  with  a  damp  cloth, 
to  avoid  charging  it  with  electricity.1 

Before  weighing  with  a  balance  the  case  should  be 
levelled  and  firmly  supported,  the  scale-pans  should 
be  scrupulously  cleaned  and  returned  to  their  places, 
and  any  dust  which  may  have  collected  on  the  knife- 
edges  or  their  bearings  should  be  cautiously  removed 
with  a  camel's-hair  brush.  The  beam  is  now  thrown 
into  operation  by  the  mechanism  already  alluded  to. 
If  the  instrument  is  correctly  adjusted,  the  pointer 
attached  to  the  under  side  of  the  beam  will  oscillate 
slowly  and  for  some  time  through  nearly  equal  arcs 
on  either  side  of  the  central  division  of  a  scale 
(/,  Fig.  15)  directly  behind  it.  If  it  tends  to  one 
side,  that  side  is  the  lighter ;  and  bits  of  paper  or 
tinfoil  should  be  fastened  to  the  scale-pan  until  an 
exact  balance  is  established.2 

In  loading  the  pans,  pincers  should  be  used  as 
much  as  possible.  In  the  case  of  the  smaller  weights, 
especially,  contact  with  the  fingers  should  be  avoided. 
It  makes  no  difference,  theoretically,  where  the  loads 
in  the  pans  are  placed  ;  but  many  practical  difficulties 
will  be  avoided  by  keeping  them  as  nearly  as  possible 
in  the  centre.  Both  loads  should  be  at  the  same 


1  By  nibbing  the  glass  at  one  side  of  a  balance   case  with  a 
piece  of  silk,  a  considerable  error  may  be  introduced  into  a  weigh- 
ing.   The  student  should  be  cautioned,  in  general,  against  the  effect 
of  charges   of  electricity  on    delicate    instruments.     An    eyeglass 
rubbed  on  the  sleeve  has  been  known  to  cause  serious  errors  in 
physical  measurement. 

2  See,  however,  first  footnote,  T  26. 


30  THE   BALANCE. 

temperature  as  the  air  within  the  balance  case  ;  for 
though  heat  weighs  nothing,  a  hot  body  may  be 
lifted  slightly  by  upward  currents  of  hot  air  around 
it.  With  non-metallic  loads  we  should  avoid  friction, 
which,  as  we  have  seen,  may  generate  charges  of 
electricity.  When  magnetic  matter  (as  iron  or  steel) 
is  to  be  weighed,  all  magnets  (§  126)  should  be  re- 
moved from  the  immediate  neighborhood.  In  an 
actual  weighing,  the  scale-pans  should  be  prevented 
from  swinging,  both  on  account  of  currents  of  air 
and  because  of  the  irregular  motion  given  to  the 
pointer. 

^[  20.  Method  of  "Weighing  by  Oscillations.  —  The 
reading  of  a  pointer  is  usually  taken  while  it  is  in 
motion,  since  much  time  would  be  lost  in  waiting  for 
it  to  come  to  rest,  and  even  then  friction  might  stop 
it  somewhat  on  one  side  of  its  true  position  of  equi- 
librium. While  in  motion  the  pointer  swings  first  to 
one  side  of  its  position  of  equilibrium,  then  to  the 
other.  The  furthest  point  reached  in  a  given  swing 
to  the  right  or  to  the  left  is  called  as  the  case  may  be 
a  right-hand  or  a  left-hand  turning-point.  Owing  to 
friction,  each  swing  is  smaller  than  the  one  before  it ; 
hence  the  position  of  equilibrium  is  not  exactly  mid- 
way between  any  two  successive  turning-points.  To 
avoid  errors  from  this  source  we  adopt  the  following 
rule  :  observe  any  ODD  l  number  of  consecutive  turning- 

1  The  object  of  making  an  odd  number  of  observations  is  that  the 
first  and  last  may  be  on  the  same  side ;  for  in  this  case  the  turning- 
points  on  one  side  are  on  the  whole  neither  earlier  nor  later  than  on 
the  other  side,  and  the  gradual  diminution  of  the  swing  affects  each 
average  alike. 


U  21. J  WEIGHING  BY  OSCILLATIONS.  31 

points  ;  find  the  average  of  those  on  the  right  and  the 
average  of  those  on  the  left ;  add  these  averages  alge- 
braically and  divide  by  2.  The  result  is  the  point 
about  which  the  oscillation  is  taking  place,  and  at 
which  the  index  tends  eventually  to  come  to  rest. 

It  is  convenient  for  many  reasons  to  call  the  middle 
scale-division  number  10,  not  0,  since  otherwise  plus 
and  minus  signs  must  be  employed.  In  practice  it 
is  sufficient  to  observe  three  consecutive  turning- 
points  of  the  index. 

It  is  frequently  impossible  to  balance  a  given  load 
exactly  by  any  combination  of  weights  which  we  are 
able  to  obtain.  Let  us  suppose  that  with  a  weight, 
w,  the  index  tends  to  rest  at  a  distance  from  the 
middle-point  equal  to  x  scale-divisions ;  while  with 
the  smallest  possible  addition  of  weight,  a,  it  tends  to 
rest  on  the  other  side  of  the  middle-point  and  at  a 
distance  from  it  equal  to  y  scale  divisions.  Then  the 
exact  weight  indicated  for  the  load,  /,  is  (see  §  41), 


The  quantity  x  +  y  is  called  the  sensitiveness  of  the 
balance  to  the  weight  (#)  under  the  load  (£) ;  and  as  it 
occurs  in  all  exact  estimations  of  weight  by  interpola- 
tion, it  may  be  made  properly  the  subject  of  further 
investigation. 

^[  21.  Determination  of  the  Sensitiveness  of  a  Bal- 
ance. —  To  test  the  sensitiveness  of  a  balance  with 
the  pans  empty,  after  carefully  adjusting  it  as  sug- 
gested in  ^T 19,  we  add  a  small  weight,  let  us  say  2  eg. 


32 


THE  BALANCE. 


[Exp.  6. 


to  the  left  hand  pan.  Instead  of  swinging  about  the 
middle  scale-division,  which  we  have  agreed  to  call 
number  10,  it  will  swing  about  a  new  point  corres- 
ponding, let  us  say,  to  number  12-6  on  the  scale.  This 
would  show  that  the  balance  is  sensitive  to  the  extent 

Of  12-6 10,  or  2-6  divisions  for  2  eg.,  or  1-3  divisions 

per  eg.,  when  the  pans  contain  little  or  no  load  besides 
their  own  weight.  This  fact  is  recorded  by  making 
a  cross  (as  in  Fig.  16)  on  a  piece  of  co-ordinate 

paper  at  the  right  of 
the  number  0,  repre- 
senting the  load,  and 
below  the  number 
(1-3)  representing  the 
sensitiveness  in  ques- 
tion. 

We  now  place,  let  us 
say,  20  grams  in  each 
pan,  and  find  as  before  the  sensitiveness  per  centi- 
gram. It  will  not  necessarily  be  the  same  as  when 
the  pans  are  empty ;  in  fact,  a  difference  is  almost 
always  observed.1  The  sensitiveness  is  then  found 
with  50  grams  in  each  pan,  and  finally  with  100  grams 
in  each  pan.  Thus,  in  an  actual  case,  a  balance  which 
was  sensitive  with  the  pans  empty  to  the  extent  of 
1-3  divisions  per  eg.,  was  affected  to  the  extent  of  1-6 
divisions  per  eg.  with  20  g.  in  each  pan,  1*4  divisions 

1  It  will  be  shown  in  1T  22  that  the  effect  of  a  load  on  the  sensitive- 
ness of  a  balance  cannot  be  anticipated ;  hence  the  student  who 
records  faithfully  what  he  sees,  not  what  he  expects  to  see,  will  here 
as  elsewhere  in  Physical  Measurement,  be  likely  to  obtain  the  most 
accurate  results.  (See  §  30.) 


/ 

0       •/        * 

•3    -4 

•j 

'    -6    -7     '8 

\ 

So 

60 
t+0 
20 
0 

\ 

\ 

x 

\ 

jXj 

\ 

^ 

"Zs 

FIG.  16. 


H  22.]  SENSITIVENESS  OF  A  BALANCE.  33 

per  eg.  with  50  g.  in  each  pan,  and  1-2  divisions  per 
eg.  with  100  g.  in  each  pan.  These  results  are  re- 
corded, as  before,  by  crosses  in  the  proper  places 
(see  Fig.  16),  and  a  curve  is  drawn  by  a  bent  ruler 
through  these  crosses.  This  curve  enables  us  to  find 
approximately  the  sensitiveness  of  the  balance  un- 
der any  ordinary  load  by  the  method  explained  in 
§  59. 

When  we  know  the  sensitiveness  (s)  of  a  balance 
to  1  eg.,  a  single  observation  of  the  pointer  is  suf- 
ficient to  determine  exactly  the  weight  indicated. 
If  w  is  the  lighter  weight  (in  the  pan  toward  which 
the  pointer  inclines)  and  x  the  number  of  scale- 
divisions  between  the  resting  point  of  the  index  and 
the  middle  of  the  scale,  the  load  (I)  indicated  is 
found  by  substituting  s  for  x  -\-  y  and  .01  for  a  in  the 
formula  of  ^[  20  ;  or 


^[  22.  Conditions  on  which  the  Sensitiveness  of  a 
Balance  Depends.  —  In  order  that  a  balance  may  move 
perceptibly  under  the  influence  of  a  very  small 
weight  added  to  either  pan,  the  central  knife-edge 
(6,  Fig.  15)  on  which  the  beam  turns  must  not  only 
be  sharp  (^[  19),  but  must  pass  nearly  through  the 
centre  of  gravity.  If  the  centre  of  gravity  is  above 
this  knife-edge,  the  balance  will  be  "top  heavy." 
This  difficulty  must  be  remedied  by  attaching  a  bit 
of  sealing-wax  to  the  pointer  below  the  knife-edge  i, 
or  by  lowering  the  centre  of  gravity  in  any  other 


34  THE   BALANCE.  [Exp.  6. 

manner.1  If  on  the  other  hand  the  centre  of  gravity 
is  too  low,  the  balance  will  be  too  steady,  and  it  will 
not  respond  sufficiently  to  a  small  change  in  the  load. 
In  this  case  it  is  necessary  to  fasten  a  small  weight 
to  the  balance  beam,  somewhere  above  the  knife- 
edge  5,  or  otherwise  to  raise  its  centre  of  gravity. 

When  the  balance-pans  are  loaded,  new  considera- 
tions come  in.  Since  in  all  positions  of  the  beam  the 
loads  hang  vertically  beneath  their  respective  knife- 
edges,  the  result  is  the  same  as  if  they  were  concen- 
trated at  those  knife-edges.  Let  us  suppose  that  the 
instrument  has  been  adjusted  so  as  to  be  sufficiently 
sensitive  when  the  pans  are  empty.  In  order  that 
it  may  remain  equally  sensitive  when  loaded,  the 
three  knife-edges  must  be  in  the  same  straight  line, 
as  m  J.,  Fig.  17.  If  the  two  outer  knife-edges  wnich 


£  -  ---  £ 

/TF7\ 


FIG.  17. 


bear  the  loads  (see  a",  c"  in  (7)  are  distinctly  above 
the  central  knife-edge  (6"),  the  combined  effect  of 
the  loads  will  be  towards  unstable  equilibrium  ;  or  if 
the  outer  knife-edges  (see  a',  c'  in  B),  are  below  the 
central  knife-edge  (&'),  the  combined  effect  of  the 
loads  will  be  to  steady  the  balance,  and  hence  to 
diminish  its  sensitiveness.  There  are  therefore  three 
types  to  which  a  balance  beam  may  belong,  repre- 

1  A  movable  screw  or  counterpoise  is  provider!  in  some  balances 
for  the  purpose  of  raising  or  lowering  the  centre  of  gravity. 


723.]  RATIO   OF   THE   BALANCE   ARMS.  35 

sented  by  the  three  diagrams,  A,  B,  and  0.  In  the 
first,  the  load  does  not  affect  the  sensitiveness,  except 
in  so  far  as  friction  may  be  concerned ;  in  the  second, 
it  lessens  it ;  in  the  third,  it  may  increase  the  sensi- 
tiveness until  the  balance  actually  becomes  u  top 
heavy." 

A  common  balance  may  belong  successively  to  all 
three  of  the  types,  (7,  A,  and  B.  Let  us  suppose  that 
with  the  pans  empty  the  extremities  of  the  beam  are 
bent  upward,  as  in  C.  With  a  medium  load,  the  beam 
may  be  straightened,  as  in  A,  and  with  a  still  greater 
load  the  ends  may  be  bent  downward,  as  in  B. 

Such  a  balance  would  be  more  sensitive  with  a 
small  load  in  each  pan  than  when  the  pans  were 
empty ;  because  a  small  load,  being  insufficient  to 
straighten  the  beam,  would  raise  its  centre  of  gravity1 
as  in  C ;  but  when  already  heavily  loaded,  so  that 
the  beam  is  bent  downward  as  in  B,  the  further 
addition  of  weight  would  lessen  its  sensitiveness. 
The  curious  shape  of  the  curve  found  in  the  last 
section  (Fig.  16),  is  thus  accounted  for. 

^[  23.  Determination  of  the  Ratio  of  the  Arms  of  a 
Balance. —  The  balance  is  now  readjusted  if  necessary 
as  in  ^[  19,  so  that  the  pointer  swings  accurately 
about  the  central  division  of  the  scale  when  the  pans 
are  empty,  and  the  100  gram  weight  is  balanced 
against  its  equivalent  as  before,  only  that  small 
weights  are  added  to  one  side  or  to  the  other  to 

1  A  balance,  though  stable  with  a  heavy  or  with  a  medium  load,  as 
well  as  when  the  pans  are  empty,  may  actually  become  "top  heavy," 
with  a  small  load  in  each  pan.  In  such  a  case,  the  centre  of  gravity 
should  be  permanently  lowered. 


36  THE   BALANCE.  [Exp.  6. 

bring  the  pointer  as  nearly  as  possible  to  the  central 
division,  and  the  exact  weight  estimated  as  in  ^[  21, 
considering  as  the  load,  Z,  that  weight  which  is  ap- 
parently the  larger.  The  loads  in  the  two  pans  are 
now  interchanged,  readjusted  by  the  use  of  the  small 
weights,  and  compared  exactly  as  before.  The  pans 
being  once  more  emptied,  the  pointer  should  swing 
about  the  central  division,  otherwise  the  balance  must 
be  readjusted  and  the  process  described  in  this  sec- 
tion must  be  repeated  until  the  equilibrium  of  the 
balance  remains  undisturbed. 

The  object  of  testing  the  balance,  as  above,  with 
equal  weights  in  the  opposite  scale-pans,  is  to  discover 
any  inequality  which  may  exist  in  the  length  of  the 
balance  arms  (a  b  and  b  c,  Fig.  17).  Such  an  inequal- 
ity might  seriously  affect  the  accuracy  of  results,  and 
we  have  no  right  to  neglect  it  even  in  ordinary  weigh- 
ings without  some  test  similar  to  the  one  described. 
It  is  true  that  by  the  method  of  double  weighing 
(see  §  44),  errors  due  to  the  inequality  of  the  balance 
arms  may  be  eliminated ;  but  double  weighings  are 
sometimes  impracticable,  as  in  the  case  of  a  body  of 
variable  weight,  or  in  a  very  long  series  of  determina- 
tions. In  such  cases  the  inequality  of  the  balance 
arms  should  be  found  by  a  careful  and  extended  series 
of  observations.  For  the  purposes  of  this  course  of 
experiments,  a  single  determination  will  suffice.  The 
ratio  of  the  balance  arms  is  calculated  therefrom  as 
explained  in  the  next  section. 

^[  24.  Calculation  of  the  Ratio  of  the  Balance  Arms. 
—  If  the  arms  of  a  balance  are  unequal,  it  is  impor- 


124.]  RATIO   OF   THE  BALANCE   ARMS.  37 

tant  to  know  from  which  arm  the  unknown  weight  is 
suspended.  To  avoid  the  necessity  of  mentioning  in 
each  case  the  pan  containing  the  load  in  question,  it 
is  customary  to  place  the  unknown  weight  at  the 
left  hand  whenever  a  single  weighing  is  to  be  made. 
In  this  way  the  known  weight,  consisting  generally 
of  several  small  pieces,  is  conveniently  adjusted  by 
the  right  hand. 

To  find  the  proportion  which  the  weight  on  the 
left  arm  always  bears  to  the  weight  on  the  right  arm, 
we  need  only  a  single  comparison  between  two  known 
weights.  As  these  weights  are  inversely  as  their 
respective  arms  (see  §  113),  the  proportion  in  ques- 
tion is  equal  to  the  ratio  of  the  right  arm  to  the  left 
arm.  Thus  if  (in  an  extreme  case)  101  grams  in  the 
left-hand  pan  balance  100  grams  in  the  right-hand 
pan,  the  right  arm  must  be  y|}^  or  1.01  times  as  long 
as  the  left  arm.  All  weights  in  the  left-hand  pan  are 
therefore  1%  greater  than  those  which  balance  them 
in  the  right-hand  pan  ;  hence  to  find  the  value  of 
an  unknown  weight  in  the  left-hand  pan  we  multiply 
that  of  the  known  weight  in  the  right-hand  pan  by 
1.01.  The  ratio  of  the  balance  arms  is  in  general  that 
number  by  which  the  known  weight  must  be  mul- 
tiplied in  order  to  find  the  unknown  weight  "which 
balances  it.  We  usually  require,  as  we  have  seen, 
the  ratio  of  the  right  arm  to  the  left  arm.  This  is 
found  by  dividing  a  known  weight  in  the  left-hand 
pan  by  a  known  weight  in  the  right-hand  pan  which 
balances  it. 

The  object  of  interchanging  the  two  weights  in 


38  THE   BALANCE.  [Exp.  7 

^f  23,  each  nominally  equal  to  100  grams,  is  to  avoid 
mistakes  arising  from  a  difference  between  the  two 
weights  in  question.  If  no  such  difference  exists, 
the  interchange  will  not  affect  the  result.  Otherwise 
to  find  the  ratio  of  the  balance  arms,  we  take  the 
average  of  the  two  weights  in  the  left-hand  pan,  and 
divide  it  by  the  average  of  the  two  weights  in  the 
right-hand  pan.  In  taking  these  averages  we  accept 
the  nominal  values  of  the  weights  in  question,  any 
errors  in  which  are  practically  eliminated  by  the 
method  of  interchange  (§  44)  here  adopted. 

EXPERIMENT  VII. 
CORRECTION    OP    WEIGHTS. 

^  25.  Process  of  Testing  a  Set  of  Weights.  —  The 
brass  1  gram  weight  is  first  balanced  against  all  the 
smaller  weights,  which  should  together  be  equal  to 
1  gram  ;  then  each  2  gram  weight  against  the  1  gram 
plus  the  smaller  weights;  then  the  5  gram  weight 
against  the  two  2  gram  weights  plus  the  1  gram  ; 
then  in.  the  same  way  the  10,  20,  50,  and  100  gram 
weights,  each  against  its  equivalent.  Whenever 
there  are  two  ways  of  making  an  equivalent,  that 
selection  is  made  by  which  the  fewest  weights  may 
be  employed.  (See  §  36,  2d  ed.)  The  100  gram 
weight  is  finally  balanced  against  a  standard.1  In 

1  The  standard  should  be  of  the  same  material  as  the  set  of  weights 
employed,  that  is,  of  brass;  but  if  any  other  material  is  used,  a  cor 
rection  must  be  made  for  the  unequal  buoyancy  of  the  atmosphere 
upon  the  loads  in  the  two  pans.  See  §  07  and  Table  21. 


U  26.]  ESTIMATION  OF   TENTHS.  39 

each  case,  where  two  weights  are  balanced,  the  differ- 
ence between  them  is  estimated  by  the  method  of 
vibration  (^[  20),  and  recorded  as  will  be  explained 
below.  To  avoid  corrections  named  in  the  last  ex- 
periment, the  method  of  double  weighing  is  used  in 
every  case. 

^f  26.  Estimation  of  Tenths  in  "Weighing.  —  In  a  long 
series  of  weighings,  as  in  testing  a  set  of  weights,  it 
is  hardly  thought  to  be  advisable  (see,  however,  §  33) 
to  record  each  turning-point  of  the  index  as  in  ^[  20. 
The  student  who  wishes  to  make  any  extended  use 
of  the  balance  should  learn  to  estimate  correctly  the 
point  of  the  scale  about  which  the  index  is  swinging, 
and  hence  the  number  of  divisions  from  the  middle 
of  the  scale *  to  the  point  where  the  index  tends  to 
rest;  to  carry  this  number  in  the  head  while  finding 
by  inspection  of  figure  16  (see  ^[  21  and  §  59)  the 
sensitiveness  of  the  balance  under  the  load  in  ques- 
tion,2 and  to  divide  mentally  the  number  thus  carried 
in  the  head  by  that  representing  the  sensitiveness  of 
the  balance,  or  the  effect  of  1  eg.  (See  general  rules 
for  interpolation,  §  41.)  He  will  thus  find  the  frac- 
tion of  a  centigram  necessary  to  make  the  index 
swing  about  the  middle-point  of  the  scale,  and  will 

1  Instead  of  adjusting  the  balance  as  in  U"  19,  so  that  the  index 
may  swing  about  the  middle-point  of  the  scale,  the  advanced  student 
may  often  prefer  to  observe  accurately  the  point  about  which  the 
index  actually  oscillates  when  the  pans  are  empty,  and  to  measure  all 
distances  from  this  point. 

2  It  is  sometimes  quicker  to  add  one  centigram  to  the  lighter  pan, 
and  thus  to  re-determine  the  sensitiveness.     In  many  cases  the  sen- 
sitiveness may  be  recalled  from  memory  with  a  sufficient  degree  of 
exactness. 


40  THE   BALANCE.  [Exr.  7. 

record  the  number  of  milligrams  nearest  to  that  frac- 
tion with  the  proper  algebraic  sign. 

Thus  if  with  a  weight  marked  10  gl  in  the  left- 
hand  pan  and  with  10  g2  in  the  right-hand  pan,  the 
index  swings  about  a  point  corresponding  to  10'3  of 
the  scale,  —  that  is,  0-3  divisions  to  the  right  of  the 
middle-point,  —  and  if  the  sensitiveness  of  the  balance 
with  a  load  of  10  grams  is  about  1-5  divisions  per 
centigram  (see  Fig.  16,  ^[  21),  the  weight  10  ^  is 
clearly  heavier  than  10  g2  by  0-3  -^  1-5  =  |  eg.  or  2 
mgr.  We  record  such  an  observation  as  follows : 
10  gi  =  10  g.2  -f  2  mgr. 

In  the  same  way  we  enter  the  result  of  placing 
10  gL  in  the  right-hand  pan  and  10  g2  in  the  left- 
hand  pan ;  and  if  there  is  any  difference,  we  find  the 
average  excess  of  10  g:  over  10  #2,  or  the  reverse. 

^[  27.  Calculation  of  the  Corrections  for  a  Set  of 
Weights.  —  Any  one  familiar  with  algebra  can  find 
the  relations  existing  between  the  different  weights 
of  a  set  from  a  series  of  equations  obtained  as  in  the 
last  section.  The  following  suggestions  may  how- 
ever be  useful.  Call  the  value  of  the  1  gram  weight 
G :  find  the  total  value  of  the  smaller  weights  (100 
eg.}  in  terms  of  this.  For  instance,  let 

100  eg.  =  G  +  1  mgr. 

Then  find  the  value  of  the  2  gram  weights,  2  gl  and 
2  ga  in  terms  of  G.     If  for  example, 
2^,  =  100  eg.  -f  G  —  1  mgr., 

we  find,  substituting  for  100  eg.  its  value,  G  + 1  mgr., 
2^  =  0  +  1  mgr.  +  G  —  1  mgr.  =  2  G; 


t27.]  CORRECTION  OF  WEIGHTS.  41 

and  if  still  further,  it  has  been  observed  that 

20,  =  2&-f  2w0r., 
we  find  similarly 

2fc=2G-f2*itfr. 

Again,  if  by  observation 


we  have 

5#=2G+2G+2  mgr.  +  G  +  1  mgr. 
=  5 


In  the  same  way  we  find  the  values  of  all  the  weights 
in  terms  of  G,  until  we  come  finally  to  the  standard. 
Knowing  the  standard  in  terms  of  G,  we  find  G  in 
terms  of  the  standard.  The  corrected  value  of  G 
should  be  expressed  in  grams  and  carried  out  to  five 
places  of  decimals.  Substituting  this  value  in  all  the 
equations,  we  obtain  finally  the  correction  in  mgr. 
for  each  weight  belonging  to  the  set  from  1  gram 
upwards. 

This  method  of  framing  and  reducing  equations  is 
not  peculiar  to  a  set  of  weights.  The  student  may 
substitute  for  it,  if  he  prefers,  the  correction  of  a  set 
of  standard  electrical  resistances,  which  he  will  learn 
how  to  compare  in  Experiment  87.  The  same  method 
may  be  applied  to  any  other  standards  capable  of 
being  arranged  like  a  set  of  weights,  so  that  each  one 
may  be  compared  with  an  equivalent  made  up  of  the 
others  below  it.  The  general  principle  by  which 
such  a  standard  set  is  corrected  is  one  of  the  best 
illustrations  of  the  method  of  multiplication  (§  39) 
upon  which  nearly  all  measurements  are  founded. 


42  THE   BALANCE.  [Exp.  8. 

EXPERIMENT  VIII. 

WEIGHING   WITH    A    BALANCE. 

^[  28.  Determination  of  Weight  in  Air  by  a  Balance. 
—  The  apparent  weight  of  a  body  in  air  may  be 
found  approximately,  as  has  been  explained  in  Ex- 
periment 1,  by  placing  it  in  one  pan  of  a  balance  — 
the  left  being  understood  unless  otherwise  stated 
(see  ^[  24)  —  and  finding  by  trial  (^[  2)  the  requisite 
number  of  weights  to  counterpoise  it.  The  accurate 
determination  of  weight  in  air  differs  from  this  rough 
method  chiefly  in  the  delicacy  of  the  instrument  em- 
ployed, and  in  the  consequent  care  of  manipulation 
(see  ^[  19).  In  this,  as  in  all  other  accurate  deter- 
minations with  the  balance,  unless  otherwise  stated, 
it  is  assumed  that  the  method  of  weighing  by  oscil- 
lations is  employed  (^[  20). 

The  object  recommended  for  this  experiment  is  a 
glass  ball,  the  weight  of  which  will  be  needed  later 
on  in  the  course.  To  prevent  it  from  rolling  out  of 
the  pan,  it  may  be  set  in  the  middle  of  a  small  ring 
of  known  weight,  which  we  will  suppose  to  be  coun- 
terpoised with  one  of  equal  weight  in  the  opposite 
pan. 

It  is  necessary  in  this  experiment  either  to  know 
the  ratio  of  the  balance  arms  (see  ^[  23),  or  to  employ 
the  method  of  double  weighing  (§  44)  as  in  Experi- 
ment 7.  The  density  of  air  must  also  be  determined 
by  an  observation  of  the  barodeik  (^[18),  or  by  an 
observation  of  the  atmospheric  pressure,  temperature, 


T  29. 


THE   HYDROSTATIC  BALANCE. 


43 


and  humidity  (*f[^[  13-15).  We  must  also  know  the 
material,  and  hence  approximately  the  densities  of 
both  the  object  weighed  and  the  weights  with  which 
it  is  counterpoised.  These  densities  may  be  found 
with  a  sufficient  degree  of  accuracy  by  referring  to 
Tables  8-11.  The  correction  of  apparent  weights  to 
vacuo  is  then  made  as  explained  in  §  68. 


EXPERIMENT  IX. 

THE   HYDROSTATIC   BALANCE,   I. 

*J[  29.  Determination  of  the  Density  of  Solids  by 
the  Hydrostatic  Balance.  —  An  arch  is  placed  over 
a  balance  pan  as  in  Fig.  18,  so  as  not  to  inter- 
fere with  its  free  vibration ;  and  on  the  middle  of 
the  arch  is  set  a  beaker.  The 
glass  ball  weighed  in  the  last  ex- 
periment is  now  bound  in  a  net- 
work of  fine  wire  and  suspended 
by  a  single  strand  from  the  hook 
of  the  balance,  so  as  to  clear  the 
bottom  of  the  beaker.  The  latter, 
being  moved  if  necessary  so  that 
its  sides  may  not  touch  the  ball,  is 
filled  with  a  quantity  of  distilled 
water  sufficient  to  cover,1  in  all 
positions  of  the  balance,  both  the 
ball  and  its  network  of  wire.  All  bubbles  of  air 

1  A  small  loop  of  wire,  projecting  above  the  surface,  may  com- 
pletely ruin  a  determination. 


FIG.  18. 


44  THE  HYDROSTATIC  BALANCE.  [Exp.  9. 

clinging  to  the  ball,  or  wire,  must  now  be  removed 
with  a  camel's-hair  brush.  The  suspending  wire, 
being  likely  to  attract  grease  or  other  foreign  matter 
which  repels  water,  is  cleaned  if  necessary,  so  that 
it  may  be  kept  wet  for  a  distance  of  about  one  centi- 
metre above  the  level  of  the  water,  by  the  continual 
oscillation  of  the  balance.  The  capillary  phenomena 
already  noticed  in  ^[  5  are  thus  reduced  to  a  small 
and  nearly  constant  amount.1 

By  these  adaptations  the  instrument  which  we 
employ  has  been  completely  transformed  into  a  "  hy- 
drostatic balance,"  by  which  the  weight  of  the  ball 
and  wire  in  water  may  now  be  found,  as  in  the  last 
experiment,  by  counterpoising  it  with  weights  in  air 
(see  Fig.  15,  ^[  19).  The  method  of  weighing  by 
oscillations  is  not,  however,  recommended  in  the  case 
of  a  hydrostatic  balance ;  but  rather  a  direct  obser- 
vation of  the  pointer  in  its  position  of  equilibrium, 
which,  owing  to  fluid  friction,  is  quickly  reached. 

Apart  from  friction,  the  sensitiveness  of  a  hydro- 
static balance  is  always  somewhat  less  than  that  of 
the  same  balance  when  used  for  measuring  weights 
in  air,2  and  must  therefore  be  re-determined  by  adding 
a  centigram  to  the  smaller  of  the  two  loads  when 
nearly  balanced  and  observing  the  result  (see  ^[  21). 
In  this,  as  in  all  experiments  with  the  hydrostatic 

1  The  use  of  spirits  of  wine  to  diminish  still  further  the  capillary 
action  (Trowbridge,  "  New  Physics,"  page  17),  is  not  recommended  to 
beginners,  on  account  of  the  danger  of  its  mixing  with  the  water  and 
thus  affecting  its  density. 

8  The  variable  amount  of  water  displaced  by  the  suspending  wire 
tends  to  increase  the  stability  of  the  balance. 


f  29.]       THE  HYDROSTATIC  BALANCE.         45 

balance,  the  temperature  of  the  liquid  should  be 
observed  both  before  and  immediately  after  finding 
the  weight  of  a  solid  in  it. 

The  weight  of  the  wire  in  water  must  be  found 
separately  in  the  same  manner  and  under  the  same 
conditions  as  before.1  The  ball  is  removed  from  the 
network  of  wire  so  as  to  leave  the  latter  undisturbed 
in  so  far  as  possible,  and  water  is  added  to  the  beaker 
in  order  that  the  same  amount  of  wire  may  be  sub- 
merged in  each  case.  It  may  even  be  necessary,  if  a 
coarse  wire  is  used,  to  adjust  the  level  of  the  water 
exactly  to  a  given  mark,  and  if  the  network  is  bulky, 
to  raise  or  lower  the.  temperature  of  the  water  to  the 
same  point  as  before. 

The  apparent  weight  of  the  ball  in  water  is  found 
by  subtraction,  and  reduced  to  vacua  by  the  principle 
of  §  67.  The  difference  between  the  apparent  weights 
in  air  and  in  water  gives  the  apparent  weight  of  wa- 
ter displaced  (§  66),  and  hence  the  volume  displaced 
(see  Table  22).  The  difference  between  the  weight 
of  the  ball  in  vacua  (^[  28)  and  its  weight  in  water 
(reduced  to  vacua  as  explained  above)  gives,  by  a 
strict  interpretation  of  the  Principle  of  Archimedes 
(§  64),  the  weight  in  vacua  of  water  displaced,  and 
hence  also  its  volume  (by  Table  23).  We  have  thus 
two  methods  of  calculating  volume,  of  which  the  first 
is  more  generally  useful,  as  it  does  not  require  any 
previous  reduction  of  weights  to  vacua;  but  the 

1  Precautions  similar  to  those  which  follow  are  necessary  when- 
ever a  method  of  difference  is  employed.  For  further  illustration 


46  THE    HYDROSTATIC  BALANCE.  [Exr.  10. 

second  is  more  rigorous,  because,  depending  upon 
weights  in  vacuo,  the  results  will  not  be  affected  by 
variations  of  apparent  weight  due  to  changes  in  at- 
mospheric density.  The  latter  should  therefore  be 
employed  when  any  considerable  time  elapses  between 
the  determinations  of  weight  in  air  and  in  water. 
The  density  (or  average  density)  of  the  ball  is  finally 
calculated  (see  ^f  1)  by  dividing  its  weight  in  vacuo 
by  its  volume.  (See  ^[  4,  also  §  68.) 


EXPERIMENT    X. 

THE   HYDROSTATIC   BALANCE,    II. 

^[  30.  Determination  of  the  Density  of  Liquids  by 
the  Hydrostatic  Balance.  —  The  experiment  consists 
essentially  of  a  repetition  of  Experiment  9,  substitu- 
ting, however,  for  distilled  water  some  other  liquid  of 
greater  or  less  buoyancy. 

Various  modifications  of  this  experiment  may  be 
necessary  according  to  the  nature  of  the  liquid  used  ; 
for  instance  in  the  case  of  strong  acids,  platinum 
wire  must  be  substituted  for  iron,  which  would  be 
speedily  dissolved,  and  even  platinum  cannot  be 
used  in  aqua  regia.  To  avoid  fumes  in  the  balance 
case,  the  suspending  wire  is  sometimes  carried  down 
through  a  series  of  small  holes  to  a  beaker  below. 
To  avoid  evaporation,  in  the  case  of  volatile  liquids, 
the  beaker  should  alwa}Ts  be  covered  with  cork  or 
cardboard  perforated  for  the  suspending  wire.  The 
same  precaution  should  be  taken  when  moisture  is 


H31.]  THE   HYDROSTATIC   BALANCE.  47 

likely  to  be  absorbed.  In  some  liquids  scarcely  any 
bubbles  are  formed ;  in  others,  such  as  glycerine,  it 
may  take  hours  to  remove  them,  though  their  forma- 
tion may  be  prevented  if  the  glycerine  is  poured  in  a 
continuous  stream  down  the  sides  of  the  beaker.  In 
most  liquids  the  effects  of  temperature  are  greater 
than  in  the  case  of  water  (see  Table  11),  hence  the 
thermometer  must  be  read  with  the  greatest  care. 
It  is  well  to  warm  or  cool  the  liquid  (and  hence  also 
the  ball)  to  the  temperature  of  the  water  in  Experi- 
ment 9,  to  avoid  all  corrections  for  temperature. 

^[31.  Calculation  of  the  Density  of  Liquids  by  the 
Hydrostatic  Method.  —  We  find  in  the  same  way  as 
in  Experiment  9,  the  apparent  weight  of  the  ball  in 
the  liquid,  allowing  for  the  wire  as  before  ;  and  from 
this  we  subtract  the  weight  of  air  displaced  by  the 
brass  weights  (see  §  67),  to  find  the  true  weight  of 
the  ball  in  the  liquid.  The  difference  between  its 
true  weight  in  the  liquid  and  that  in  vacua,  already 
found  (^[  28),  is  equal  to  the  weight  in  vaciw  of  the 
liquid  displaced.  This  follows  from  the  Principle  of 
Archimedes  (§  64). 

The  volume  of  liquid  displaced  is  of  course  equal 
to  the  volume  of  the  ball,  which  will  not  differ  per- 
ceptibly from  the  value  previously  determined  (see 
end  of  ^[  29)  if  the  temperatures  of  the  two  experi- 
ments are  nearly  the  same.  If  this  is  not  the  case, 
it  is  necessary  to  allow  for  an  expansion  or  contrac- 
tion of  the  glass,  at  the  rate  of  about  one  part  in 
40,000  for  every  degree  Centigrade.  (See  Table  8  b 
and  §  83.) 


48  THE   HYDROSTATIC   BALANCE.  [Exp.  10. 

The  weight  in  vacua  of  the  liquid  displaced  is 
finally  divided  by  its  volume  to  find  its  density. 

The  weight  in  vacua  may  be  checked  by  calculating 
the  apparent  weight  of  the  liquid  displaced,  as  in  Ex- 
periment 9,  then  reducing  at  once  to  weight  in  vacua 
by  applying  the  necessary  factor  from  Table  21,  as 
explained  in  §  68,  using  the  density  already  calcu- 
lated. This  latter  method  is  slightly  inaccurate,  as 
has  been  stated  before  (^[  29),  on  account  of  its  dis- 
regarding variations  of  atmospheric  density  during 
the  course  of  experiments. 

In  determining  the  density  of  water  by  the  hydro- 
static balance,  the  weight  displaced  may  be  found 
as  in  Experiment  9  or  10 ;  but  the  volume  displaced 
cannot  be  calculated  in  the  manner  explained  above, 
because  the  tables  which  we  employ  themselves  de- 
pend upon  the  density  of  water.  It  is  necessary  to 
calculate  the  volume  of  the  solid  immersed  from 
actual  measurements  of  its  dimensions1  (see  ^[  1). 
By  this  method,  essentially,  with  the  aid  of  instru- 
ments of  precision,  accurate  determinations  of  the 
density  of  water  have  been  made  (see  Table  25). 
The  student  will  have  an  opportunity  in  Experiment 
19,  to  confirm  these  determinations  within  the  limit 
of  accuracy  of  the  instruments  which  he  employs. 

1  The  volume,  »,  of  the  glass  ball  may  be  calculated  from  its 
diameter,  d,  by  the  formula,  v  =  -5236  d3.  In  place  of  the  glass  ball 
we  may  use,  for  purposes  of  illustration,  the  rectangular  block  whose 
volume  has  already  been  determined  in  Experiment  1.  If  it  floats 
in  water,  a  lead  sinker  may  be  attached  to  it.  The  sinker  must 
remain  in  place  after  the  block  is  removed,  in  order  that  its  weight 
may  be  allowed  for.  A  spring  balance  maj'  be  used  to  find  roughly 
the  weight  of  water  displaced.  See  Exercises  7-10  in  the  Descriptive 
list  of  Experiments  in  Physics  published  by  Harvard  University. 


-  32.)  CAPACITY  OF  VESSELS.  49 

EXPERIMENT   XI. 

CAPACITY   OF   VESSELS. 

^[  32.  Determination  of  the  Capacity  of  a  Specific 
Gravity  Bottle.  —  Any  bottle  with  a  solid  stopper  of 
ground-glass  may  be  used  for  finding  the  specific 
gravity  of  liquids ;  but  when  solids  are  to  be  intro- 
duced, one  with  a  wide  mouth  will  be  needed.  The 
capacity  of  the  bottle  is  determined  in  the  following 
manner.  The  bottle  is  first  washed  in  perfectly  pure 
water,  then  dried  with  a  cloth  inside  and  out,  and 
afterwards  still  more  thoroughly  dried  with  a  hot  air- 
blast.1  The  weight  of  the  bottle  is  found  within  a 
centigram,  then  the  bottle  is  alternately  dried  and 
weighed  until  by  the  agreement  of  two  successive 
weighings,  the  drying  is  known  to  be  complete.  The 
last  weight  found,  if  confirmed  by  the  method  of 
double  weighing  as  in  ^[  28,  is  the  apparent  weight 
of  the  bottle  in  air.  It  is  understood  that  the  stop- 
per is  always  weighed  with  the  bottle.  In  this  case,  it 
should  be  placed  in  the  scale-pan  beside  the  bottle, 
so  that  the  density  of  the  air  may  be  the  same  inside 
and  out.  The  bottle,  which  will  be  warmed  by  the 
hot  air-blast,  must  be  allowed  time  to  cool  to  the 
temperature  of  the  room  before  the  weighing  is 
completed,  since  otherwise  currents  of  hot  air  might 
seriously  affect  the  result  (see  ^[  19). 

1  When  a  hot  air-blast  cannot  be  had,  the  bottle  may  be  dried  by 
rinsing  it  out  several   times  with  a  small  quantity  of  alcohol,  and 
exposing  it  for  a  few  minutes  to  a  draught  of  air. 
4 


50  SPECIFIC   GRAVITY  BOTTLE.  [Exp.il. 

The  bottle  is  then  filled  with  distilled  water  at  an 
observed  temperature,  not  far  from  that  of  the  room ; 
then  closed  in  such  a  manner  (see  Fig.  19)  as  to 
allow  all  bubbles  of  air  to  escape.1  The  outside  of 
the  bottle  is  then  carefully  dried  with  a  cloth  or 
blotting-paper.  The  weight  is  again 
found  with  the  same  degree  of  ac- 
curacy as  before,  and  immediately 
afterward  the  temperature  of  the 
water  and  the  density  of  the  air 

The  difference  between  the  two 
apparent  weights  of  the  bottle  con- 
taining air  and  water,  respectively, 
is  equal  to  the  apparent  weight  in 
air  of  the  water  which  it  contains  (§  66) ;  this 
weight  of  water  multiplied  by  the  space  occupied  (at 
the  higher  of  the  two  observed  temperatures,  see 
^[  33)  by  a  quantity  of  water  weighing  apparently 
1  gram  (in  air  of  ttte  observed  density,  see  Table  22), 
gives  the  total  space  occupied  by  the  water,  or  in 
other  words  the  capacity  of  the  bottle  at  the  observed 
temperature. 

^[  33.  Effects  of  Varying  Temperature  on  a  Specific 
Gravity  Bottle.  —  It  is  hardly  necessary,  in  the  experi- 
ments which  follow,  to  allow  for  the  expansion  of 
the  glass  bottle  due  to  changes  of  temperature  which 

1  If  the  shape  of  the  stopper  makes  this  impossible,  it  must  be 
altered  by  grinding  or  by  filling  up  any  hollows  in  it  with  paraffine  or 
other  material  not  acted  upon  by  ordinary  liquids.  In  this  case  the 
weight  in  air  must  be  re-determined. 


133.]  EFFECTS  OF  TEMPERATURE.  51 

it  is  likely  to  undergo.1  In  a  laboratory,  maintained 
as  it  should  be  at  a  nearly  constant  temperature, 
these  changes  will  be  slight.  Unless,  however,  spe- 
cial precautions  are  taken  to  keep  the  water  in  the 
bottle  at  a  constant  temperature,  serious  errors  are 
likely  to  arise.  These  errors  will  be  still  greater  in 
the  case  of  certain  other  liquids  which  we  shall  em- 
ploy. The  expansion  of  alcohol,  for  instance,  will 
be  found  to  be  several  hundred  times  as  great  as  that 
of  glass  (see  Table  11). 

Let  us  first  suppose  that  the  liquid  which  fills  a 
closed  bottle  is  gradually  cooling,  and  hence  in  the 
process  of  contraction.  A  bubble  will  soon  be  formed. 
This  need  not,  however,  give  rise  to  apprehension  if 
the  initial  temperature  (at  which  the  bottle  was  filled) 
has  been  correctly  observed;  for  the  weight  of  the 
liquid  will  not  be  changed  by  its  contraction,  and  the 
bubble  weighs  practically  nothing.  We  may  there- 
fore determine  the  weight  of  a  liquid  which  fills  a 
bottle  a,t  an  observed  temperature,  after  it  has  fallen 
below  that  temperature. 

Now,  let  us  suppose  that  the  liquid  is  growing 
warmer;  and  hence,  expanding,  that  it  forces  its 
way  out  by  the  stopper,  yet  clings  to  the  bottle. 
Unless  the  liquid  is  volatile  or  hygroscopic,2  its  weight 

1  The  capacity  of  a  vessel  increases  by  the  same  amount  as  the 
volume  of  a  solid  of  the  same  material  which  would  exactly  fill  the 
vessel.     In  the  case  of  glass,  this  increase  is  at  the  rate  of  about  1 
part  in  40,000  per  degree  Centigrade. 

2  Hygroscopic  liquids,  such  as  sulphuric  acid  or  chloride  of  cal- 
cium, should  be  slightly  warmed  before  the  experiment,  so  that  they 
may  be  weighed  while  cooling. 


52  SPECIFIC  GRAVITY  BOTTLE.  [Exp.  11. 

will  be  unchanged,  and  hence  may  be  determined  at 
leisure.  If,  however,  the  liquid  evaporates  immedi- 
ately (as  many  liquids  do)  on  contact  with  the  air, 
there  will  be  a  continual  loss  of  weight.  In  such 
cases,  we  must  find  the  temperature  as  nearly  as  pos- 
sible at  the  time  of  weighing,  when  it  will  be  seen 
that  the  quantity  of  liquid  weighed  exactly  fills  the 
bottle. 

In  practice,  both  the  initial  and  final  temperatures 
are  usually  observed ;  the  former  just  before  the  in- 
sertion of  the  stopper,  the  latter  immediately  after 
completing  the  weighing.  We  notice  that  with  a 
non-volatile  liquid,  the  initial  temperature  is  always 
required;  and  the  same  statement  applies  to  a  volatile 
liquid  which  is  cooling ;  but  with  a  volatile  liquid  in 
general  it  is  the  maximum  temperature  which  we 
wish  to  determine.  In  no  case  do  we  take  the  mean 
of  the  two  temperatures  before  and  after  the  ex- 
periment. 

The  liquids  which  we  employ  should  be  warmed 
or  cooled  if  necessary,  so  that  they  may  be  nearly  at 
the  same  temperature  as  the  room ;  since  otherwise 
the  rapid  changes  of  temperature  which  must  ensue 
(§  89)  would  make  an  accurate  observation  of  the 
thermometer  impossible.  Errors  in  weighing  might 
also  be  introduced,  owing  to  currents  of  hot  or  cold 
air  (^[  19).  In  the  case  of  certain  liquids  (as  ether) 
which  are  apt  to  become  cold  through  evaporation,1 

1  Care  must  be  taken  in  general  to  prevent  evaporation;  and 
especially  in  the  case  of  impure  liquids,  tlie  strength  of  which  would 
be  affected  by  the  escape  of  the  more  volatile  ingredients. 


f  34.  J  DISPLACEMENT.  53 

there  is  danger  that  moisture  may  be  condensed  on 
the  sides  of  the  containing  vessel  (see  ^[  17).  Par- 
ticular care  must  be  taken  in  the  case  of  water,  when 
below  the  temperature  of  the  room ;  lest  through  the 
humidity  of  the  air  or  from  other  causes  it  should 
fail  to  evaporate  as  fast  as  it  is  driven  out  of  the 
bottle.  Any  moisture  collected  around  the  stopper 
should  be  removed  with  blotting-paper  before  making 
a  final  adjustment  of  the  weights. 


EXPERIMENT    XII. 

DISPLACEMENT     I. 

^[  34.  Determination  of  Displacement  by  the  Specific 
Gravity  Bottle.  —  The  experiment  consists  essentially 
of  a  repetition  of  Experiment  11,  with  a  bottle  al- 
ready partly  full  of  sand,  or  any  other  substance 
insoluble  in  water.  The  capacity  of  the  bottle  for 
water  is  evidently  less  than  before  by  an  amount 
exactly  equal  to  the  space  which  the  sand  takes  up ; 
hence  the  latter  can  be  found  by  subtracting  the 
new  capacity  from  the  old.  This  method  of  deter- 
mining volume  is  especially  convenient  in  the  case 
of  powders,  which  cannot  easily  be  suspended  from 
a  hydrostatic  balance. 

Certain  modifications  of  the  methods  used  in 
Experiment  11  are  introduced  when  finely  divided 
substances  are  employed.  Even  with  sand  consid- 
erable difficulty  may  be  found  in  removing  the 
bubbles  of  air  which  cling  to  it  under  water.  By 


54  SPECIFIC  GKAVITY  BOTTLE.  [Exp.  12 

continual  shaking  with  water  in  a  well-stoppered 
bottle,  this  air  may  generally  be  freed  from  the  sand.1 
To  obtain  dry  sand,  it  should  be  heated  before  the 
experiment  to  a  temperature  above  100°. 

The  same  process  may  be  used  to  dry  various 
powders  not  easily  melted  or  decomposed  by  heat; 
but  others  require  special  precautions  belonging  to 
the  province  of  Chemistry  rather  than  Physics. 

It  may  be  observed  that  the  apparent  weight  of 
the  solid  used  in  this  experiment  is  incidentally 
determined ;  for  we  have  only  to  subtract  from  the 
apparent  weight  of  the  bottle  with  it  that  of  the 
bottle  without  it  as  found  in  the  last  experiment. 
The  density  of  the  solid  may  therefore  be  calculated 
as  in  Experiment  9. 

^[  35.  Illustration  of  the  Principle  of  Archimedes,  — 
To  understand  what  is  meant  by  the  water  displaced 
by  a  solid,  the  bottle  may  be  filled  with  water  as  in 
Experiment  11,  then  the  solid  may  be  introduced  ; 
water  will  be  literally  displaced,  and  if  the  whole 
quantity  thus  driven  out  of  the  bottle  could  be  col- 
lected and  weighed,  we  should  have  a  direct  measure- 
ment of  the  water  displaced  lay  the  solid.  In 
practice  we  prefer  to  find  this  by  difference. 

If  we  call  s  the  apparent  weight  of  the  sand,  b  that 
of  the  bottle,  w  that  of  the  water  which  fills  it,  and 
d  that  of  the  water  displaced  by  the  sand,  the  weights 
observed  are  (1)  b  and  (2)  b  -\-  w  in  Experiment  11, 

1  An  air-pump  greatly  facilitates  the  process,  but  unless  special 
precautions  are  taken  the  water  is  apt  to  bubble  over  into  the  re- 
ceiver and  to  find  its  way  into  the  valves  of  th  pump. 


f  35.]  DISPLACEMENT.  55 

(3)  b  +  s  and  (4)  b  -f  s  +  w>  —  d  in  Experiment  12. 
The  apparent  weight  of  water  which  fills  the  bottle 
is  the  difference  between  the  first  and  second  obser- 
vations, or  (2)  —  (1),  but  when  the  sand  is  already  in 
the  bottle  the  quantity  of  water  required  is  the 
difference  between  the  last  two  observations,  or  (4) 
—  (3)  ;  hence  the  quantity  displaced  is  [(2)  —  (1)J 

-[(4)  —  (3)]. 

Now  the  weight  of  the  sand  in  air  is  evidently  the 
difference  between  the  first  and  third  observations,  or 
(3)  —  (1)  ;  its  apparent  weight  in  water  is  the  dif- 
ference between  the  second  and  fourth,1  or  (4)  — 
(2)  ;  its  loss  of  weight  in  water  is  therefore  [(3)  — 
(1)]  —  [(4)  —  (2)].  This  is  seen  by  comparison  to 
be  identical  with  the  expression  above  for  the  weight 
of  water  displaced. 

The  student  who  finds  difficulty  in  realizing  how 
the  apparent  weight  or  loss  of  weight  of  a  solid  in 
water  can  be  found  by  the  specific  gravity  bottle 
may  repeat  these  measurements  with  a  hydrostatic 
balance,  using  a  cup  to  hold  the  sand  in  place  of  the 
network  of  wire  employed  in  Experiment  9  to  hold 
the  glass  ball ;  or  he  may  find  the  weight  and  loss  of 
weight  in  water  of  the  steel  balls  used  in  Experiment 
4  by  means  of  the  specific  gravity  bottle.  The  Prin- 
ciple of  Archimedes  (§  64)  states  that  loss  of  weight 

1  In  both  observations  we  have  the  same  weight  of  the  bottle, 
and  the  same  hj'drostatic  pressure  of  the  water  upon  the  bottom 
or  sides  of  the  bottle  (§  63)  ;  the  only  difference  is  the  downward 
pressure  of  the  sand,  which  is  present  in  (4)  and  absent  in  (2).  This 
pressure  exerted  under  water  is  what  we  call  the  weight  of  the 
sand  in  water. 


56  SPECIFIC   GRAVITY  BOTTLE.  [Exp.  13. 

in  water  (which  we  think  of  as  determined  by 
hydrostatic  methods)  is  equal  to  the  weight  of  water 
displaced  (which  we  think  of  as  determined  by  a 
specific  gravity  bottle).  The  agreement  of  the  results 
obtained  by  hydrostatic  methods  with  those  from 
the  specific  gravity  bottle  may  serve  therefore  either 
as  an  illustration  of  this  principle  or  as  a  mutual 
confirmation  of  these  results. 


EXPERIMENT  XIII. 

DISPLACEMENT   II. 

If  36.  Determination  of  the  Volume  and  Density  of 
Solids  Soluble  in  Water.  When  owing  to  the  solubility 
in  water  of  the  substance  employed,  the  method  ex- 
plained in  the  last  experiment  cannot  be  applied,  it 
remains  only  to  find  some  other  fluid  of  known  den- 
sity in  which  that  substance  is  insoluble.  The  vari- 
ous products  of  the  distillation  of  petroleum  are 
especially  suited  to  this  purpose,  since  they  dissolve 
few  (if  any)  ordinary  substances  which  are  soluble 
in  water.  We  may  occasionally,  with  great  care,  use 
a  saturated  aqueous  solution  of  the  substance  whose 
density  is  to  be  determined,  or  a  liquid  which  has 
been  allowed  to  act  chemically  upon  an  "excess" 
of  that  substance,  since  in  either  of  these  cases  the 
liquid  will  have  no  further  action  on  the  solid.  Gases 
may  also  be  employed ;  but  on  account  of  the  diffi- 
culty of  measuring  their  weight  correctly  even  by  the 
most  delicate  balances,  it  is  customary  to  estimate 


T37.]  SPECIFIC   VOLUMES.  57 

the  quantity  present  by  a  direct  or  indirect  measure- 
ment of  its  volume.1  Owing,  however,  to  the  ten- 
dency of  certain  substances  to  absorb  large  quantities 
of  gas,  all  such  methods  may  lead  to  erroneous  and 
even  absurd  results. 

For  sake  of  simplicity  we  will  choose  the  liquid 
whose  density  has  been  determined  in  Experiment 
10,  and  for  the  solid  some  substance  insoluble  in  that 
liquid ;  and  in  order  that  the  density  of  the  liquid 
may  be  the  same  as  before,  it  should  be  warmed  or 
cooled  if  necessary  to  the  temperature  observed  in  Ex- 
periment 10.  With  such  a  solid  and  liquid,  Experi- 
ment 12  is  to  be  essentially  repeated. 

^[  37.  Calculation  of  Volume  and  Density  by  the  Use 
of  Specific  Volumes.  We  have  already  seen  how  the 
weight  of  water  displaced  by  a  solid  may  be  found 
either  by  the  hydrostatic  balance  (Experiment  9) 
or  by  the  specific  gravity  bottle  (Experiment  12). 
By  the  same  methods  we  may  obtain  the  weight  of 
any  other  fluid  displaced  by  a  solid.  We  have  al- 
ready applied  this  principle  in  Experiment  10  for 
determining  the  density,  of  a  liquid.  Knowing  the 
weight  in  grams  and  the  number  of  cubic  centimetres 
displaced,  we  found  by  division  the  weight  of  1  cu. 
cm.  It  would  have  been  equally  simple  to  inter- 
change the  divisor  and  dividend,  and  thus  to  find  the 
space  in  cu.  cm.  occupied  by  1  gram.  This  is  some- 
times called  the  specific  volume  of  a  liquid. 

The  mutual  relations  existing  between  the  weight 

1  For  a  description  of  the  "  Volumenometer,"  see  Trowbridge's 
New  Physics,  Experiment  31. 


58  SPECIFIC  GRAVITY  BOTTLE.  [Exp.  14. 

w,  the  volume  v,  the  density  c?,  and  the  specific  vol- 
ume s,  of  any  substance  are  given  by  the  equations 

.  s  =  -,  v  —  w  s,  etc. 
d 

The  specific  volume  is  therefore  technically  the  "  re- 
ciprocal "  of  the  density.  To  find  it  we  divide  unity 
by  the  density  already  determined  in  Experiment  10, 
or  by  that  which  we  may  find  from  Experiment  14. 

We  have  already  used  specific  volumes  in  Table 
23  (see  1"  29),  and  we  know  that  the  weight  in  vacua 
of  the  liquid  displaced,  multiplied  by  its  specific  vol- 
ume,1 gives  the  actual  volume  displaced,  which  is  of 
course  equal  to  that  of  the  solid  causing  the  displace- 
ment. The  volume  of  the  solid  enables  us  to  reduce 
its  apparent  weight  to  vacuo  (§  67),  and  hence  to  cal- 
culate its  density  (§  68). 


EXPERIMENT  XIV. 

DENSITY   OP   LIQUIDS. 

IF  38.  Determination  of  the  Density  of  a  Liquid  by  the 
Specific  Gravity  Bottle.  We  have  already  found  the 
weight  of  a  bottle  containing  water  and  air,  and  we 
have  calculated  its  capacity ;  it  remains  only  to  find 
its  weight  when  filled  with  any  other  fluid,  in  order 

1  The  student  should  bear  in  mind  that  the  specific  volume  here 
employed  is  the  space  occupied  by  a  quantity  of  liquid  weighing  1 
gram  in  vacuo,  not  that  which  weighs  apparently  1  gram  in  air.  True 
specific  volumes  must  be  multiplied  by  true  weights  in  vacuo  to  find 
actual  volumes.  Apparent  specific  volumes  (see  Table  22)  are  in- 
tended to  give  the  same  result  with  apparent  weights  in  air. 


139.]  THE   DENSIMETER.  59 

that  the  density  of  that  fluid  may  be  determined. 
For  the  purpose  of  comparison  we  will  choose  the 
liquid  already  used  in  Experiments  10  and  13,  and 
warm  or  cool  it,  as  nearly  as  may  be  convenient,  to  the 
temperature  of  those  experiments.  The  actual  tem- 
perature should  be  observed  for  reasons  explained  in 
If  38,  both  before  and  immediately  after  weighing. 
The  barodeik  should  also  be  read,  in  order  to  make 
sure  that  no  great  change  has  taken  place  in  the 
course  of  our  experiments  with  the  specific  gravity 
bottle,  since  otherwise  its  apparent  weight  in  air  must 
be  re-determined. 

The  apparent  weight  of  a  quantity  of  alcohol  suffi- 
cient to  fill  the  bottle  is  found  by  subtracting  that  of 
the  bottle  with  air  from  that  of  the  bottle  filled  with 
alcohol,  and  is  reduced  to  vacua  as  explained  in  §  67. 
The  density  is  then  calculated  by  dividing  the  weight 
in  vacua  by  the  capacity  of  the  bottle,  from  IT  32. 
The  strength  of  the  alcohol  is  finally  found  by  ref- 
erence to  Table  27,  using  a  process  of  double  inter- 
polation (see  §  58).  The  strength  of  the  alcohol 
may  also  be  calculated  from  the  data  of  Experiment 
14 ;  and  even  if  the  temperatures  in  Experiments  10 
and  14  differ  considerably,  the  two  results  should 
agree  in  respect  to  strength. 

EXPERIMENT  XV. 

THE   DENSIMETER. 

^[  39.  Hydrometers  and  Densimeters.  —  There  are 
various  kinds  of  hydrometers  employed  in  the  arts. 


60  DENSITY  OF  LIQUIDS.  [Exp.  15. 

Nicholson's  has  been  already  described,  and  is  the 
type  of  a  "hydrometer  of  constant  immersion  ;"  that 
is,  one  which  in  use  is  always  made  to  sink  in  a 
liquid  to  a  given  mark.  A  common  glass  hydrometer 
is,  on  the  other  hand,  an  example  of  "variable  immer- 
sion." The  distance  it  sinks  in  a  fluid  depends  upon 
the  density  of  the  fluid,  and  is  read  by  a  scale  at- 
tached to  the  stem  of  the  instrument.  The  scales 
used  in  the  arts  are  generally  arbitrary.  The  principal 
ones  are  those  invented  by  Baume',  Beck,  Cartier,  and 
Twaddell,  which  are  compared  in  Table  40  with  a 
scale  of  density.  The  instruments  most  convenient 
for  scientific  purposes  carry  a  scale  which  indicates 
at  once  the  density  of  the  liquid,  and 
hence  bear  the  name  of  densimeters. 

The  sensitiveness  of  a  densimeter 
evidently  depends  upon  the  smallness 
of  the  graduated  stem,  compared  with 
the  whole  displacement  of  the  instru- 
ment ;  but  if  we  make  the  stem  too 
small,  a  single  hydrometer  of  the  ordi- 
nary length  can  cover  only  a  very  lim- 
ited range  of  densities.  A  set  of  three 
instruments  is  often  used, — one  for  liq- 
uids lighter  than  water,  one  for  liquids 
heavier  than  water,  and  one  for  liquids 
of  intermediate  density.  There  are  also 
sets  of  twelve  or  more  hydrometers, 
covering  together  the  whole  range  of 
densities  from  sulphuric  acid  (1.8)  to  ether  (0.7). 
With  these  great  accuracy  and  rapidity  may  be 


140.]  THE   DENSIMETER.  61 

attained,  even  without  applying  any  of  the  ordinary 
corrections ; 1  but  if  rapidity  be  the  chief  object,  a 
single  instrument  with  a  "  specific  gravity  scale"  will 
be  found  most  convenient.  Such  a  one  is  often  called 
by  dealers  a  "  Universal  hydrometer  "  (see  Fig.  20). 

The  errors  of  such  instruments  are  not  so  great  as 
one  might  expect,  considering  that  the  scales  are 
printed  in  quantities  from  originals  none  too  care- 
fully made,  fitted  to  tubes  of  by  no  means  uniform 
bore,  regardless  within  certain  limits  of  their  size, 
and  fastened  to  these  tubes  at  a  point  too  high  or 
too  low,  as  the  case  may  be.  Still,  even  if  the  read- 
ing in  water  is  found  to  be  nearly  correct,  considerable 
errors  may  be  discovered  in  other  parts  of  the  scale. 
As  these  errors  depend  largely  upon  the  calibre  of 
the  tube,  the  process  of  correcting  them  may  be 
properly  called  calibration  (  §  36). 

^[  40.  Calibration  and  Use  of  a  Densimeter.  —  The 
reading  of  the  instrument  is  taken  while  floating 
successively  in  at  least  three  standard  liquids  of 
known  density,  such  as  water,  alcohol,  and  glycerine 
(see  Tables  25-27),  then  in  a  number  of  other  liquids 
whose  density  is  to  be  determined.  As  with  a  Nich- 
olson's hydrometer,  the  under  surface  of  the  liquid  is 
(when  possible)  used  as  a  sight  (see  Fig.  6,  ^f  6)  ; 
and  the  same  precautions  are  taken  to  avoid  friction 
against  the  sides  of  the  jar,  and  the  effects  of  capil- 

1  It  should  be  remembered  that  changes  of  atmospheric  density 
influence  only  that  portion  of  a  hydrometer  which  is  above  the 
liquid,  and  hence  will  not  generally  affect  even  the  fourth  place  of 
decimals.  The  effect  of  a  narrow  range  of  temperature  in  changing 
the  volume  of  a  glass  hydrometer  is  equally  unimportant. 


62 


DENSITY   OF   LIQUIDS. 


[Exp.  15. 


lary  action  due  to  the  stem's  becoming  dry  near  the 
surface  of  the  liquid.  Both  the  densimeter  and  the 
thermometer  (which  is  invariably  read  in  every  ob- 
servation) must  be  washed  after  immersion  in  each 
liquid,  either  under  the  faucet  or  in  three  changes  of 
water;  they  should  also  be  carefully  dried  before 
immersion  iu  a  new  liquid ;  otherwise  more  or  less 
dilution  or  mixture  is  sure  to  take  place.  The  cor- 
rections of  the  densimeter  are  then  calculated  and 
applied  as  explained  in  the  next  section. 

^[  41.  Treatment  of  Corrections  by  the  Graphical 
Method.  —  Correction  and  error  are  by  definition 
(§  24)  equal  and  opposite.  ]f  the  observed  value 
of  a  quantity  is  greater  than  its  real  value,  we  say 
that  the  error  is  positive,  the  correction  negative. 
Thus,  by  subtracting  the  observed  from  the  tabulated 
densities  of  water,  alcohol,  and  glycerine  at  a  given 
temperature,  we  find  the  several  corrections  for  the 
instrument  by  which  these  densities  were  observed. 

The  correction  of  an 
instrument  will  gen- 
erally vary  according 
to  the  reading  in 
question  ;  hence,  to 
find  the  correction 
for  every  reading,  it 
is  necessary  to  con- 
struct either  a  table 

of  corrections  or  a  curve.  Thus,  in  Fig.  21  the  three 
points  indicated  by  crosses  represent  (see  §  59)  cor- 
rections of  a  particular  densimeter  corresponding  to 


c 

•02 
01 
CC 
•01 
<B 

•*   o 

?    ft?     1-0    l-l    1-2    1-3    14  /•& 

^^ 

^r- 

" 

—> 

• 

1 

BALANCING  COLUMNS. 


three  densities  :  namely,  for  alcohol,  density  0.80, 
correction  — .004  ;  for  water,  density  1.00,  correc- 
tion —  .002 ;  for  glycerine,  density  1.25,  correction 
-{-.004.  The  curve  drawn  by  a  bent  ruler  through 
the  crosses  enables  us  to  find  approximately  the  cor- 
rection of  this  instrument  for  all  intermediate  densi- 
ties by  the  general  rules  of  the  graphical  method 
(§  59).  Thus  for  an  ammoniacal  solution  of  the 
density  0.9  or  thereabouts,  the  correction  would  be 
not  far  from  — .003.  Corresponding  corrections  should 
be  applied  to  each  of  the  liquids  whose  density  has 
been  determined  by  means  of  the  densimeter. 


EXPERIMENT  XVI. 

BALANCING    COLUMNS. 

If  42.  Determination  of  Density  by 
Methods  of  Balancing  Columns.  The 
ordinary  method  of  balancing  col- 
umns is  illustrated  in  Figure  22. 
Some  mercury,  for  instance,  is  poured 
into  a  U-tube,  then  into  the  longer 
arm  some  water.  Suppose  the  mer- 
cury is  thus  forced  up  to  a  level,  5, 
in  the  shorter  arm,  and  down  to  a 
level,  <?,  in  the  longer  arm,  by  a  col- 
umn of  water  reaching  from  a  to  <?, 
and  let  the  vertical  distances,  be  and 
ac,  between  the  corresponding  levels 
be  measured  ;  then  since  the  den- 
sity of  water  is  known,  the  density 


a— 


FIG. 


64  DENSITY  OF  LIQUIDS.  [Exp.  16. 

of  mercury  will  be  determined  (see  IT  43).  When  the 
two  liquids  are  miscible  this  method  cannot  be  applied. 
Another  method  in  which  this  difficulty  is  avoided 
is  illustrated  in  Figure  23.  A  tube  in  the  form  of  an 
inverted  Y  is  plunged  into  two  ves- 
sels, c,  containing  water,  and  d,  let  us 
say,  glycerine.  The  two  liquids  are 
then  sucked  up  cautiously  to  the  re- 
spective levels  a  and  b;  and  held 
there  by  closing  a  stop-cock  in  the 
stem  of  the  Y.  The  relative  density 
of  the  glycerine  will  then  be  deter- 
mined by  measuring  the  distances  ac 
and  bd.  These  distances  are  meas- 
ured vertically,  in  the  case  of  each 
liquid,  between  its  level  in  the  tube 
and  its  level  in  the  cistern. 

For  measuring  long   distances,   as 
ac  or  bd,  a  millimetre  scale   behind 
FIG.  23.  the  tubes  will  suffice  ;  for  short  dis- 

tance (as  be,  Fig.  22)  a  vernier  gauge  may  be  prefer- 
able;  but  special  care  must  be  taken  to  have  the 
shaft  vertical.  To  diminish  the  effects  of  capillary 
attraction,  the  tubes  should  have  a  diameter  of  a 
centimetre  if  possible,1  and  the  level  should  be  read 

1  If  smaller  tubes  are  used,  two  experiments  must  be  made.  In 
one,  the  columns  of  liquid  should  be  as  long  as  may  be  convenient; 
in  the  other  as  short  as  possible.  The  effects  of  capillary  action  are 
then  eliminated  in  the  usual  manner  by  taking  differences  (see  §  32). 
Thus  instead  of  the  column  ab  (Fig.  22)  we  find  the  difference  be- 
tween two  such  columns  in  the  two  experiments ;  and  in  the  same 
way  we  find  the  difference  between  the  two  columns  be.  These  dif- 
ferences evidently  balance  one  another. 


U43.1  BALANCING  COLUMNS.  65 

by  the  middle  point  of  the  surface,  whether  convex 
or  concave  (see  case  of  the  Barometer,  If  13). 

The  common  temperature  of  the  two  balancing 
columns  may  be  found  by  a  mercurial  thermometer 
midway  between  them.  An  observation  of  the  baro- 
deik  will  be  unnecessary. 

IF  43.  Theory  of  Balancing  Columns.  Two  liquid  col- 
umns are  said  to  balance  one  another  when  they  exert 
equal  and  opposite  pressures  at  a  given  point.  Since 
pressure  is  affected  by  the  density  as  well  as  by  the 
depth  of  a  fluid,  the  greater  height  of  one  column 
must  counterbalance  the  greater  density  of  the  other. 
In  other  words,  the  densities  of  two  balancing  col- 
umns must  be  to  each  other  inversely  as  their  vertical 
heights. 

It  is  evident  that,  in  Figure  22  of  the  last  article, 
the  vertical  height  of  the  water  is  equal  to  ac,  the 
total  length  of  the  column  ;  but  that  of  the  mercury 
which  balances  it  is  only  a  portion  of  the  whole  col- 
umn of  mercury,  namely,  be  ;  for  the  part  in  the  bend 
of  the  tube  having  the  same  level,  c,  at  both  ends, 
exerts  no  pressure  to  the  right  or  to  the  left  (§  62), 
and  serves  simply  to  transmit  pressure  from  one  col- 
umn to  the  other.  For  the  same  reason,  we  disre- 
gard in  Figure  23  the  portions  of  the  liquids  below 
c  and  d,  and  find  that  the  balancing  columns  are 
ac  and  bd. 

The  balance  between  the  two  liquid  columns  in  the 

first  method  (Fig.  22)  will  not  be  disturbed  by  the 

atmospheric  pressure,  provided  that  it  affects  both 

columns  alike,  as  is  very  nearly  the  case  ;  but  strictly 

5 


66  DENSITY   OF  LIQUIDS.  [Exp.  16. 

we  must  observe  that  the  barometric  pressure  is 
greater  at  b  than  at  a.  There  is,  as  it  were,  a  column 
of  air  of  the  height  ab  acting  on  the  mercury  without 
any  equivalent  acting  on  the  water.  Since  the  den- 
sity of  air  is  about  800  times  less  than  that  of  water, 
we  should  subtract  from  the  apparent  length  of  the 
column  of  water,  ac,  one  800th  part  of  the  distance 
ai,  to  find  the  column  of  water  which  would  balance 
the  mercury  in  vacua. 

In  the  second  method  (Fig.  23),  supposing  c  and  d 
to  be  on  the  same  level,  we  find  in  the  same  way  an 
unbalanced  column  of  air,  ab,  acting  on  the  shorter  of 
the  two  columns  of  liquid.  If  the  longer  column  is  as 
before,  water,  we  subtract  from  it  one  800th  of  ab.  If 
the  shorter  is  water  we  add  one  800th  of  ab.  In  ap- 
plying this  correction,  we  neglect  the  fact  that  the  air 
within  the  tube  is  slightly  rarefied,  since  the  accuracy 
of  the  instrument  employed  will  not  justify  more  than 
a  rough  approximation  to  the  density  of  the  air  in 
question. 

If  in  either  method  I  is  the  length  of  the  column  of 
liquid  whose  density,  Z>,  is  to  be  determined,  w  the 
length  of  the  column  of  water  which  balances  it  in 
vacuo,  and  d  the  density  of  this  water  at  the  observed 
temperature  (see  Table  25),  we  have,  solving  the 
inverse  proportion  mentioned  above, 


. 


IT  44.] 


DENSITY  OF  GASES. 


67 


EXPERIMENT  XVII. 

DENSITY   OP   AIR. 

^[  44.  Determination  of  the  Density  of  Air.  —  A 
stout  flask  provided  with  a  stop-cock  (Fig.  24)  is 
made  thoroughly  dry  (see  ^[  32),  and  weighed  with 
the  stop-cock  open.  The  flask  is  then 
connected  with  an  air-pump,  and  as 
much  air  as  possible  is  exhausted. 
The  stop-cock  is  now  closed ;  and  the 
flask,  having  been  disconnected  from 
the  air-pump,  is  re-weighed.  It  should 
be  left  on  the  balance  long  enough  to 
prove  that  there  is  no  perceptible  gain 
of  weight  from  leakage  of  air  into  it, 
then  quickly  opened  under  water  as 
in  Fig.  25.  The  stop-cock  is  closed  by  some  mechan- 
ical contrivance  while  the  flask  is 
still  completely  submerged ;  then  the 
flask  is  dried  outside  and  weighed 
with  the  water  which  has  entered. 
The  temperature  of  the  water  is 
now  observed.  Finally  the  flask  is 
rilled  completely  with  water  and 
re-weighed.  When  all  these  obser- 
vations have  been  recorded,  an  ob- 
servation of  the  barodeik  (see  ^[  18) 
is  made  for  purposes  of  comparison. 
Having  found  the  proportion  of  air 
exhausted,  we  calculate  its  density,  as  explained  below. 


FIG.  24. 


FIG.  25. 


68  DENSITY  OF  GASES.  [Exp.  17. 

^[  45.  Theory  of  the  Partial  Vacuum.  —  When  a 
flask  from  which  the  air  has  been  partially  exhausted 
is  opened  under  water  as  in  Figure  25,  the  water  is 
forced  inwards  until  the  residual  air  is  sufficiently 
compressed  to  resist  the  atmospheric  pressure  from 
outside.  If  the  temperature  is  constant,  as  will  be 
essentially  the  case  when  the  flask  is  surrounded  by 
water,  the  pressure  depends  chiefly  on  the  density 
(see  §  78) ;  hence  the  residual  air  is  compressed  until 
its  density  is  the  same  as  that  of  the  outside  air. 
The  space  which  it  then  occupies,  compared  with  the 
whole  capacity  of  the  flask,  will  then  represent  the 
proportion  of  air  remaining  in  it ;  and  the  amount  of 
water  which  enters  compared  with  the  total  amount 
necessary  to  fill  the  flask  will  represent  the  proportion 
of  air  exhausted. 

The  flask  must  not  be  plunged  too  deep  below  the 
surface  of  the  water,  for  if  it  is  the  air  within  it  may 
be  perceptibly  compressed  ;  but  it  is  well  to  submerge 
it  to  a  depth  of  10  or  20  cm.,  to  offset  the  expansion 
of  the  air  caused  by  its  taking  up  vapor  from  the 
water  with  which  it  comes  in  contact  (see  Table  13). 
The  less  air  there  is,  the  less  will  be  its  expansion.  To 
obtain  accurate  results,  we  must  therefore  exhaust 
nearly  all  the  air,  or  else  substitute  for  water  some 
less  volatile  fluid. 

It  may  be  observed  that  the  water  which  enters  the 
flask  replaces,  bulk  for  bulk,  that  portion  of  the  air 
which  has  been  exhausted.  The  weight  of  this  air  is 
the  difference  between  the  weights  of  the  flask  before 
and  after  exhaustion  ;  the  weight  of  the  equivalent 


IT  46.]  DENSITY  OF  GASES.  69 

bulk  of  water  is  the  difference  between  the  last  two 
weighings,  —  before  and  after  the  admission  of  water. 
We  notice  that  in  this  experiment,  unlike  those  which 
precede  it,  the  water  enters  the  flask  without  dis- 
placing any  air  whatever;  hence  no  allowance  is 
made  for  the  weight  of  air  displaced.  Both  the 
weight  of  air  exhausted  and  that  of  the  water  which 
takes  its  place  are  affected  by  the  buoyancy  of  the 
atmosphere  upon  the  brass  weights  (§  65),  and  in 
the  same  proportion  ;  hence  their  quotient  is  unaf- 
fected, and  represents  the  true  specific  gravity  of  the 
air  referred  to  the  water.  This  should  agree  closely l 
with  the  atmospheric  density  indicated  by  the 
barodeik. 

EXPERIMENT   XVIII. 

DENSITY   OP   GASES. 

^[  46.  Determination  of  the  Density  of  a  Gas.  —  A 
light  flask,  as  large  as  the  balance  pans  will  admit,  is 
made  perfectly  dry  (see  ^[  32),  and  weighed  with  its 
stopper  beside  it.  To  determine  the  density  of  the 
air  within  the  flask,  an  observation  of  the  barodeik 
is  made  (see  ^[  18).  Then  the  flask  is  filled  with 
coal-gas  conducted  through  a  rubber  tube  reaching 
as  far  as  possible  into  the  flask.  To  prevent  the 
escape  of  the  coal-gas,  which  is  lighter  than  air,  the 

1  The  true  specific  gravity  of  any  substance  referred  to  water  at 
any  temperature  must  strictly  be  multiplied  by  the  density  of  water 
at  that  temperature  (see  §  69),  to  find  the  density  of  the  substance  in 
question.  In  the  present  case,  the  multiplication  will  hardly  affect 
the  last  significant  figure  of  the  result. 


70  DENSITY  OF  GASES.  [Exp.  18. 

flask  is  held  iu  an  inverted  position  throughout  the 
process ;  after  which  the  tube  is  drawn  slowly  out 
of  the  flask  without  checking  the  flow  of  gas  (see 
6,  Fig.  26),  and  the  stopper  (a)  is 
immediately  inserted.  The  weight  of 
the  flask  is  again  determined.  More 
gas  is  then  passed  into  the  flask  as 
before  until  it  reaches  a  constant 
weight.  The  temperature  of  the  gas 
in  the  flask  is  then  found  by  a  ther- 
mometer inserted  through  a  bored 
stopper ;  and  the  pressure  is  deter- 
mined by  an  observation  of  the  ba- 
rometer. Finally  the  flask  is  filled  with  water  and 
weighed  for  the  purpose  of  finding  its  capacity. 

The  last  weighing  and  the  observation  of  tempera- 
ture which  should  accompany  it  may  be  comparatively 
rough;  but  the  weighings  with  air  and  with  gas 
should  be  made  with  the  utmost  precision,  since  the 
difference  between  them,  upon  which  the  result 
depends,  is  so  slight  that  even  a  small  error  would 
affect  this  result  in  a  very  considerable  proportion 
(see  §  36).  If  ordinary  prescription  scales  are  used, 
tne  result  should  depend  upon  the  mean  of  at  least 
five  double  weighings  in  each  case.  When  great 
accuracy  is  desired,  a  counterpoise  should  be  used 
consisting  of  a  second  flask,  hermetically  sealed,  equal 
to  the  first  in  volume  and  nearly  equal  in  weight. 
Small  weights  added  to  the  counterpoise  should  bring 
about  an  exact  adjustment.  By  using  such  a  counter- 
poise, changes  in  atmospheric  density  are  eliminated, 


11  47.]  STANDARD  OF  LENGTH.  71 

since  the  air  will  buoy  up  the  contents  of  both  pans 
alike. 

The  capacity  of  the  flask  is  then  calculated  as  in 
^[  32,  and  the  density  of  coal-gas  at  the  observed 
temperature  and  pressure  is  found  by  the  formula  of 
§  70,  using  the  density  of  the  air  indicated  by  the 
barodeik.  The  result  is  then  reduced  to  0°  and  76 
em.  pressure  by  the  formula  of  §  81. 


EXPERIMENT  XIX. 

MEASUREMENT   OF   LENGTH. 

Tf  47.  Selection  of  a  Standard  of  Length.  —  A  care- 
ful comparison  of  the  various  scales  which  we  have 
hitherto  employed  for  the  measurement  of  length 
will  generally  show  cases  of  disagreement.  These 
may  sometimes  be  explained  as  the  result  of  expan- 
sion by  heat  (see  Table  8  i)  ;  for,  though  a  scale 
should  be  correct  at  0°,  unless  otherwise  stated,  there 
is  no  agreement  to  this  effect  among  manufacturers.1 
In  other  cases  errors  are  discovered  which  may  be 
traced  to  the  machine  by  which  the  scales  are  di- 
vided. It  will  not  do  to  assume  that  the  most 
carefully  finished  scales  are  the  most  accurate.  Those 
printed  in  large  quantities  on  wood  compare  very 

1  English  measures  are  generally  adjusted  (if  at  all)  to  a  tempera- 
ture of  about  62°  Fahrenheit.  Certain  French  manufacturers  main- 
tain that  all  standards  are  supposed  to  be  correct  at  4°  Centigrade. 
In  the  case  of  brass  metre  scales,  discrepancies  of  nearly  half  a 
millimetre  may  sometimes  be  traced  to  the  temperatures  at  which 
they  have  been  adjusted. 


72  MEASUREMENT  OF  LENGTH.  [Exp.  19. 

favorably  with  common  varieties  of  "  vernier  gauge  " 
(see  Fig.  27).  The  latter,  in  particular,  need  to  be 
tested  as  will  be  explained  below.  For  this  pur- 
pose, "  end  standards  "  are  made  by  various  manufac- 
turers with  a  considerable  degree  of  precision.  In 
place  of  these,  however,  the  student  will  find  it  more 
instructive  to  use  one  depending,  as  follows,  upon  his 
own  measurements. 

The  volume,  v,  of  a  glass  ball  has  already  been 
determined  (^[  29)  ;  from  this  the  diameter,  d,  may 
be  calculated  by  geometry,  using  the  formula J 

d  =  1.2407  #v. 

In  calculating  the  diameter  of  a  sphere  from  the 
cube  root  of  its  volume,  great  accuracy  may  be 
obtained  (see  §  36).  Thus  if  the  volume  is  really 
40.00  cu.  cm.,  and  owing  to  an  error  of  1  eg.  in  weigh- 
ing, the  observed  value  is  40.01  cu.  cm.,  the  calculated 
diameter  will  be  4.2435  cm,  instead  of  4.2432  cm. 
The  difference  (.0003  cwi.)  between  the  calculated  and 
the  true  value  would  be  imperceptible. 

If  the  ball  which  we  .employ  is  not  perfectly 
spherical,  an  average  diameter  will  be  given  by  the 
formula.  We  shall  see  in  ^[  50,  I.  how  slight  ir- 
regularities can  be  allowed  for.  We  may  therefore 
obtain  from  our  experiments  in  hydrostatics  a  stan- 
dard, in  the  form  of  a  sphere,  by  which  it  is  possible 
to  correct  the  reading  of  a  vernier  gauge,  or  any 
other  kind  of  caliper. 

1  This  is  derived  from  the  ordinary  formula  — 


T48.]  TESTING  CALIPERS.  73 

1"  48.  Testing  Calipers.  A  caliper  is  an  instrument 
intended  especially  to  determine  by  contact  the 
diameter  of  bodies,  generally  the  outside  diameter. 
It  is  provided  with  two  points  called  "teeth"  or 
"  jaws,"  one  of  which  at  least  is  movable.  In  one  class 
of  calipers  the  jaws  are  hinged  together,  their  motion 
being  magnified  in  some  cases  by  a  long  index ;  in 
another  class  there  is  a  sliding  motion,  as  in  the 
vernier  gauge  used  in  Experiment  1  (see  Fig.  27)  ; 
in  a  third  class  the  motion  is  produced  by  a  screw, 
as  in  the  micrometer  gauge  (Fig.  28). 

The  instrumental  errors  (§  31)  likely  to  arise  dif- 
fer, of  course,  according  to  the  special  construction 
of  the  gauge  in  ques- 
tion ;  but  there  are 
certain  classes  of  de- 
fects common  to  all 
calipers,  and  hence  it 
is  well,  before  begin- 
ning any  series  of 
measurements,  to 
make  a  regular  ex- 
amination of  each  in- 
strument, covering  the 
,  ,.  .  .  ,  FIG.  28. 

following  points:  — 

(a)  DISTORTION.  The  shank  of 
a  vernier  gauge  (ad,  Fig.  27)  should  appear  per- 
fectly straight  to  the  eye,  when  "  sighted "  in  the 
ordinary  manner,  and  perfectly  free  from  twist.  A 
micrometer  screw  (cd,  Fig.  28)  should  similarly  ap- 
pear straight,  so  that  the  tooth  c  may  be  .accurately 
centred  in  all  positions. 


MEASUREMENT  OF  LENGTH.  [Exp.  19. 


(5)  CONTACT.     The  jaws  of  a  gauge  must  be  able 
to  touch  each  other  at  some  point  (as  ppr  Fig.  29) 
convenient  for  measurement.     The 
shape  of  these  jaws  may  be  modi- 
fied, if  necessary,  by  the  use  of  a 
file,  or  by  the  application  of  solder, 
FIG.  29.  in  order  that  this  condition  may  be 

fulfilled.    The  location  of  the  point  of  contact  is  found 
by  examining  the  streak  of  light  between  the  jaws. 

(c)  PERPENDICULARITY.    The  surfaces  of  the  teeth 
or  jaws  at  the  point  of  contact  should  be  at  right 
angles  with  the  shank  of  the 

gauge.  In  the  case  of  a  microm- 
eter,  any  obliquity  immediately 
appears  when  the  screw  is  ro- 
tated. To  detect  it  in  a  sliding 
gauge  it  is  necessary  to  reverse 
one  of  the  jaws  (as  5  in  Fig« 
30),  and  to  see  whether  the  two 
inner  surfaces  remain  parallel. 

(d)  GRADUATION.    The  uniformity  of  the  thread 
of  a  micrometer  screw  is  sufficiently  established  if  it 
turns  in  the  nut,  when  well  oiled,  with  equal  facility 
throughout  its  entire  length.     The  graduation  of  a 
vernier  gauge  is  most  easily  tested  by  the  vernier  it- 
self ;  for  if  the  latter  always  subtends  exactly  the 
same  number  of  divisions  on  the  main  scale,  these 
may  be  assumed  to  be  sensibly  uniform. 

(e)  LOOSENESS.      A    gauge   should    slide   freely 
from  one  position  to  another ;  but  any  looseness  in 
the  moving  parts  must  be  prevented.     For  this  pur- 


FJG.  30. 


T49.]  PRECAUTIONS  IN  THE   USE   OF   CALIPERS.          75 

pose  a  set  screw  (a,  Fig.  27)  is  usually  attached  to  a 
vernier  scale.  In  the  absence  of  any  equivalent  ar- 
rangement, a  nut  may  often  be  tightened  successfully 
by  pinching  it  slightly  in  a  vice. 

If  the  defects  here  mentioned  cannot  be  over- 
come, the  caliper  should  be  discarded  for  the  purposes 
of  the  exact  measurements  which  follow. 

IT  49.   Precautions  in  the  Use   of  Calipers. 

(a)  WARMTH.  In  ordinary  measurements  with  a 
vernier  gauge,  the  warmth  of  the  hand  will  hardly 
cause  a  perceptible  expansion  ;  but  with  micrometers, 
considerable  care  must  be  taken  to  avoid  errors  from 
this  source.  The  usual  method  is  to  hold  the  instru- 
ment with  a  cloth,  but  it  is  still  more  effective  to 
mount  it  in  a  vice,  and  thus  to  leave  both  hands  free 
for  making  the  necessary  adjustments. 

(5)  CLAMPING.  When  a  caliper  has  been  "  set " 
on  a  given  object,  it  is  customary  to  clamp  it  before 
making  a  reading,  lest  in  the  mean  time  dislocation 
should  take  place.  There  is  danger,  however,  that  in 
the  very  act  of  clamping  any  instrument,  its  "  set- 
ting" may  be  disturbed.  Vernier  gauges,  unless 
specially  provided  with  springs  to  keep  the  moving 
parts  in  place,  are  troublesome  in  this  respect.  The 
difficulty  is  lessened  by  keeping  a  moderate  pressure 
on  the  clamp  while  the  setting  is  taking  place.  In  all 
instruments,  the  accuracy  of  a  setting  should  be 
tested  after  clamping. 

(<?)  STRAIN.  The  teeth  or  jaws  of  a  caliper  must 
obviously  not  be  bent  forcibly  apart  by  the  pressure 
between  them  and  the  object  on  which  they  are  set ; 


70  MEASUREMENT  OF  LENGTH.  [Exp.  19. 

for  the  bending  will  introduce  an  error  in  the  read- 
ing. One  may  judge  whether  the  pressure  is  ex- 
cessive or  not  by  the  muscular  force  required  to 
produce  it,  or  by  the  hold  which  the  caliper  seems  to 
have  upon  the  object  in  question.  The  best  microm- 
eters are  provided  with  a  friction  head  (/,  Fig.  28) 
which  slips  when  the  required  pressure  is  obtained. 
A  most  important  result  is  thus  secured,  namely,  a 
uniform  pressure  in  all  settings  of  the  gauge,  includ- 
ing the  zero  reading  (see  §  32)  whereby  the  effects  of 
strain  may  be  eliminated. 

(d)  ROUGHNESS.     If  the  surfaces  of  the  teeth  or 
jaws  of  a  caliper  are  not  perfectly  smooth  and  flat, 

an  object  may  fit  between  them 
with  greater  facility  in  some 
places  than  in  others.  To  elim- 
inate the  effects  of  any  such  ir- 
regularity, the  diameter  which 
is  to  be  measured  should  ter- 
minate in  the  points  (p  and  p' ', 
FIG.  31.  Fig>  31)  which  determine  the 

zero  reading  of  the  gauge  (see  Fig.  29). 

These  are  generally  the  most  prominent  points  of 
the  inner  surfaces  ;  hence  the  rule,  place  the  object  to 
be  measured  where  it  fits  with  the  greatest  difficulty. 

(e)  OBLIQUITY.     The  line  pp'  (Fig.  31)  is  neces- 
sarily parallel  to  the  shank  of  the  gauge ;  hence  also 
the  diameter  of  any  object  which  coincides  with  it. 
If,  however,  through  any  mistake  in  the  above  adjust- 
ment, the  diameter  to  be  measured  is   perceptibly 
inclined  with  respect  to  the  line  pp\  a  considerable 


IT  50.]  CORRECTION  OF   CALIPERS.  77 

error  is  likely  to  be  introduced  into  the  result.  It 
may  be  shown  by  trigonometry  that  if  the  inclina- 
tion is  less  than  1°,  the  error  will  be  less  than  one 
six-thousandth  part  of  the  quantity  measured ;  and 
hence  practically  insensible.  Since  the  eye  can  de- 
tect under  favorable  circumstances  an  obliquity  even 
less  than  1°,  the  following  rule  will  be  found  suffi- 
ciently accurate  :  make  the  diameter  to  be  measured 
sensibly  parallel  to  the  shank  of  the  gauge. 

(/)  POSITION.  An  object  may  be  fitted  between 
the  teeth  of  a  caliper  in  various  ways,  and  care  must 
be  taken  that  the  diameter  thus  measured  is  the  one 
sought.  In  the  case  of  a  rectangular  block,  for  in- 
stance, a  minimum  diameter  is  usually  required,  and 
care  must  be  taken  not  to  place  it  corner  wise ;  in  the 
case  of  a  sphere,  however,  a  maximum  measurement 
is  wanted,  and  to  secure  this,  especially  when  the 
teeth  are  rounded  (as  in  Fig.  32),  many  trials  must 
be  made  and  with  the  greatest  care. 

(g)  PARALLAX.  Errors  of  parallax  (§  25) 
may  be  avoided  when  two  scales  are 
mutually  inclined,  by  holding  the  eye  or 
the  gauge  in  such  a  position  that  the  lines 

appear  parallel,   as   in  A,  Fig.  33,  not  in- 
FIG.  32.       ,.       ,         •      T-J 
clmed  as  m  B. 

^[    50.    Correction  of  Calipers.  •  i      i  , 

I  j  iifli  ml'     In M|  i  in  1 1 
—  It  is  important  to  determine  |<"'T""l    /""W'l 

the  reading  of  a  gauge  or  caliper  FIG.  33. 

when   the  jaws  are  in  contact 
(see  Fig.  29).     This  is  called  the  "zero  reading," 
because  it  corresponds  to  a  distance  zero  between  the 


78  MEASUREMENT  OF  LENGTH.  [Exp.  19. 

points  p  and  p'  where  contact  takes  place.  A  gauge 
need  not  be  condemned  simply  because  the  "  zero 
reading  "  is  not  exactly  zero.  The  fulfilment  of  this 
condition  is  in  fact  exceedingly  rare.  It  is  only  neces- 
sary that  the  zero  reading  shall  be  accurately  deter- 
mined, in  order  to  avoid  (by  subtraction)  all  errors 
from  this  source  (§  32). 

I.  VERNIER  GAUGE.  The  general  method  of  reading 
a  vernier  gauge  has  been  explained  in  ^[  3.  We  have 
seen  in  §  37  how  the  tenths  of  the  millimetre  divisions 
on  the  main  scale  are  read  by  means  of  a  "  vernier." 

In  case,  however,  the  indication  of  the  vernier  lies 
between  two  numbers,  it  becomes  necessary  in  all 
exact  measurements  to  estimate  fractions  of  tenths. 
We  have  already  found  a  rough  way  of  representing 
such  fractions  (see  ^T  6).  A  more  exact  method  is 
described  in  §  37.  To  obtain  success  in  applying  this 
method  to  a  vernier  reading  to  tenths  of  a  millimetre, 
the  rulings  of  the  scale  should  be  fine,  and  a  hand 
lens  (such  as  is  represented  in  Fig.  34)  should  be 
used  to  magnify  the  ver- 
nier and  main  scale  divi- 
sions so  that  the  difference 
between  them  may  be 
plainly  visible  to  the  eye. 
The  student  will  find  it 

difficult,  at  first,  to  select  the  diagram  in  §  37  most  re- 
sembling the  case  of  coincidence  in  question  ;J  but  with 

1  One  of  the  chief  difficulties  in  conducting  this  experiment  lies  in 
the  tendency  of  students  to  hold  a  gauge  more  or  less  obliquely,  so 
that  all  cases  of  coincidence  may  appear  to  be  exact,  or  (what  is 
nearly  as  hopeless)  precisely  alike.  To  an  accurate  observer,  no  two 
settings  present  in  general  exactly  the  same  appearance. 


If  50,  L] 


THE  VERNIER   GAUGE. 


79 


a  little  practice  most  of  his  errors  should  be  confined 
to  a  range  of  one  or  two  hundredths  of  a  millimetre. 

If  the  zero  of  the  vernier  comes  opposite  a  point 
below  the  zero  of  the  main  scale,  the  reading  is 
negative.  For  convenience,  however,  the  negative 
sign  is  applied  (as  in  logarithms)  only  to  the  whole 
number  indicated  on  the  main  scale,  —  the  fraction 
remaining  positive.  Thus  if  the  zero  on  the  vernier 
passes  the  zero  on  the  main  scale  by  .02  mm.  when 
the  jaws  are  brought  into  contact,  the  reading  of  the 
vernier  should  be  .98  ;  and  in  this  case,  the  zero  read- 
ing is  1.98,  according  to  the  general  rule  given  in  ^[  3. 

When  the  zero  reading  has  thus  been  found  within 
one  or  two  huudredths  of  a  millimetre,  a  body  of 


FIG.  36. 

known  diameter  is  set  between  the  jaws  of  the  gauge. 
The  glass  ball,  for  instance,  used  in  Experiments  8 
and  9  is  to  be  placed  (see  Fig.  31),  so  as  to  reach 
between  the  points  p  and  p'  by  which  the  zero 
reading  was  determined  (see  Fig.  29).  Looking  at 
the  jaws  endwise,  we  should  see  the  ball  symmetri- 
cally situated,  as  in  Fig.  35. 

If  the  ball  is  not  perfectly  round,  we  shall  need  at 
least  10   measurements   of  its  diameter;   and  these 


80  MEASUREMENT  OF  LENGTH.  [Exp.  19. 

measurements  should  obviously  be  distributed  as  uni- 
formly as  possible  over  the  surface  of  the  sphere. 
The  student  will  do  well  to  mark  in  ink  ten  points 
upon  the  ball  as  in  B,  Fig.  36,  which  are  to  be  brought 
successively  under  the  point  p  (Fig.  35),  in  one  jaw 
of  the  gauge.  After  each  measurement,  the  corre- 
sponding mark  should  be  erased,  to  prevent  confusion. 
As  to  the  manner  of  spacing  the  ten  points  in  ques- 
tion, the  student  is  advised  to  begin  with  a  20-sided 
paper  weight  (A,  Fig.  36),  to  place  a  number  in  the 
middle  of  each  of  the  ten  faces  visible  from  a  given 
point  of  view,  then  to  copy  these  marks  on  the  glass 
ball  J5,  so  that  they  may  appear  to  be  spaced  in  the 
same  manner  in  both  cases.  The  geometrician  will 
observe  that  there  is  one  way  and  only  one  way  of 
distributing  ten  diameters  uniformly  over  the  sur- 
face of  a  sphere,  and  that  this  way  has  been  here 
practically  adopted. 

In  each  of  the  ten  measurements,  a  reading  is 
made  to  hundredths  of  millimetres ;  then  the  zero 
reading  is  re-determined.  From  the  mean  of  the  ten 
measurements  above,  the  mean  zero  reading  is  sub- 
tracted. We  thus  find  the  average  diameter  of  the 
ball  according  to  the  gauge.  Dividing  this  observed 
diameter  by  that  obtained  by  the  hydrostatic  method 
(which  we  will  suppose  to  be  the  true  diameter — see 
^[  47),  we  obtain  an  important  factor,  namely,  the 
average  space  in  millimetres  occupied  by«each  milli- 
metre division  in  a  certain  part  of  the  gauge.  If  the 
gauge  is  uniformly  graduated  (see  ^[  48,  d),  it  is  ob- 
viously possible  to  correct  all  measurements  made 


1J50,  II.]  THE  MICROMETER  GAUGE.  81 

with  the  gauge  at  the  same  temperature  by  means  of 
the  factor  thus  found.  In  practice,  however,  it  may 
be  assumed  that  a  gauge  has  been  selected  in  which 
these  corrections  are  too  small  to  be  considered. 

II.  MICROMETER  GAUGE.  —  In  place  of  the  glass 
ball  of  Experiments  8  and  9,  the  student  may  use 
the  steel  balls  of  Experiments  3  and  4,  provided  that 
the  displacement  of  these  balls  has  been  confirmed 
by  the  specific  gravity  bottle,  as  suggested  in  *fi  35. 
The  joint  volume  of  these  balls  is  then  found  by  the 
use  of  Table  22  (see  ^[  29),  then  the  average  volume, 
from  which  (the  balls  being  uniform  in  size)  the  av- 
erage diameter  is  calculated  by  the  formula  of  ^[  47. 

The  diameters  of  these  balls  may  now  be  measured 
by  means  of  a  micrometer  gauge  (see  Fig.  28).  The 
tests  to  be  applied  to  a  micrometer  and  the  precau- 
tions to  be  followed  are  essentially  the  same  as  with 
any  other  kind  of  caliper  (see  ^[  48  and  ^[  49).  The 
zero  reading  is  found  as  in  the  case  of  a 
vernier  gauge  by  bringing  the  teeth  into 
contact.  Then  the  teeth  are  separated  by 
turning  the  head  of  the  screw  (Fig.  28) 
until  the  ball  whose  diameter  is  to  be 
measured  fits  symmetrically  between  these 
teeth  as  in  Figure  37.  FlG"  3L 

The  whole  number  of  revolutions  of  the  screw 
should  correspond  with  the  number  of  main  scale 
divisions  on  the  nut  d,  uncovered  by  the  barrel  e. 
The  hundredths  of  a  turn  maybe  read  by  the  gradua- 
tion on  the  edge  of  the  barrel,  using  as  an  index  a 
mark  running  along  the  nut.  Care  must  be  taken 
6 


82  MEASUREMENT  OF  LENGTH.  [Exr.  19. 

to  avoid  a  mistake  of  a  whole  turn  in  reading  the 
gauge ;  if,  for  instance,  nine  whole  divisions  (nearly) 
are  uncovered  by  the  barrel,  and  the  index  points 
to  98  hundredths,  the  reading  is  8.98  (not  9.98).  It 
is  safer  with  many  micrometers  to  confirm  the  whole 
number  of  revolutions  by  actually  counting  them. 

In  reading  the  micrometer  the  divisions  correspond- 
ing to  hundredths  of  a  revolution  should  be  divided 
into  tenths  by  the  eye  (§  26).  A  micrometer  with 
a  millimetre  thread  thus  indicates  the  thousandth 
part  of  a  millimetre.  In  the  case  of  a  negative  zero 
reading,  as  with  the  vernier  gauge,  the  minus  sign 
should  be  applied  only  to  the  whole  number  of  turns. 

The  diameter  of  each  of  the  steel  balls  is  deter- 
mined in  this  way  to  thousandths  of  a  turn  of  the 
screw ;  and  from  the  average  reading  we  subtract 
the  average  zero  reading,  observed  before  and  after 
the  above  with  an  equal  degree  of  precision.  We 
find  in  this  way  the  average  number  of  turns  and 
thousandths  of  a  turn  actually  made  by  the  screw. 
Dividing  the  average  diameter  of  the  balls  (from  the 
hydrostatic  method)  by  the  corresponding  number  of 
turns  of  the  screw,  we  have  finally  the  distance 
through  which  the  micrometer  screw  advances  in 
each  revolution.  This  is  called  the  "pitch  of  the 
screw."  We  shall  assume  that  a  micrometer  has 
been  found,  reading  to  millimetres  and  thousandths 
so  accurately  that  in  the  case  of  objects  of  small 
diameter,  no  correction  need  be  applied. 


I  51.]        ZERO   READING  OF  A  SPHEROMETER.  83 

EXPERIMENT  XX. 

TESTING   A   SPHEROMETER. 

IT  51.  Determination  of  the  Zero  Reading  of  a  Spher- 
ometer.  A  spherometer  (Fig.  38)  is  essentially  a  mi- 
crometer (see  IT  50,  II.)  supported 
by  three  legs  (d,  /,  #).  The  verti- 
cal screw  (ce)  has  a  head  (5)  di- 
vided into  a  hundred  parts,  the 
tenths  of  which  may  be  estimated 
by  the  eye  (§  26).  The  thou- 
sandths of  a  revolution  may  thus 
be  read  by  means  of  an  index  (a). 
This  index  carries  a  vertical  scale 
(a/),  on  which  the  head  of  the  FlG  38> 

micrometer  (6)  registers  the  whole  number  of  revo- 
lutions made  by  the  screw.  Both  on  the  scale  («/") 
and  on  the  micrometer,  the  indications  should  in- 
crease as  the  screw  is  raised.  It  is  well  to  renumber 
the  main  scale  if  necessary,  so  that  negative  readings 
may  be  avoided. 

The  zero  reading  of  a  spherometer  is  its  reading 
when  the  point  of  the  central  screw  is  in  the  plane  of 
the  three  feet.  To  find  it,  the  instrument  is  set  on  a 
piece  of  plate  glass  (Fig.  39)  of  sensibly  uniform  thick- 
ness, selected  by  the  aid  of  a  micrometer  gauge,  and  the 
screw  of  the  spherometer  is  raised  or  lowered  until 
all  four  points  seem  to  touch  the  glass  at  the  same 
time  (see  Fig.  40).  If  the  central  screw  is  driven 
too  far  forward,  the  instrument  will  not  stand  firmly 


84  THE   SPHEROMETER.  [Exp.  20. 

upon  the  glass,  but  will  have  a  tendency  to  rock. 
This  will  be  noticed  especially  if  one  of  the  feet  be 
held  down  by  the  finger,  while  the  other  two  feet  are 
subjected  to  an  alternating  pressure.  In  fact,  the  con- 
ditions upon  which  rocking  depends  are  so  delicate 
that  a  change  of  a  thousandth  of  a  millimetre  may 
cause  it  to  appear  or  to  disappear.  When  the  instru- 
ment has  been  adjusted  so  that  rocking  is  barely  per- 
ceptible, the  reading  is  estimated  in  millimetres  to 
three  places  of  decimals,  in  the  same  manner  as  in 
the  case  of  a  micrometer  gauge. 


FIG  39.  FIG.  40. 

On  account  of  possible  irregularities  in  the  glass, 
at  least  five  readings  should  be  taken  in  different 
parts  of  one  surface ;  and  as  plate  glass  is  apt  to  warp 
slightly  in  the  process  of  manufacture,  five  more 
readings  should  be  taken  on  the  other  surface.  The 
mean  of  the  values  thus  found  on  a  piece  of  glass 
of  uniform  thickness  gives  the  zero  reading  of  the 
spherometer,  and  should  be  determined  after  as  well 
as  before  any  series  of  measurements  such  as  will  be 
described  in  the  next  section,  in  order  to  avoid  errors 
due  to  change  of  temperature  and  to  the  wearing 
away  of  the  points  upon  which  the  instrument  rests. 


T  53.]  PITCH  OF  SCREW.  85 

Tf  52.  Determination  of  the  Pitch  of  the  Screw.  A 
spherometer  with  a  screw  of  known  pitch  can  be  used 
in  place  of  a  micrometer  to  measure  the  diameter  of 
small  objects.  These  are  placed 
upon  the  plate  glass  already  used  to 
determine  the  zero  reading,  and  the 
screw  is  adjusted  so  as  to  touch 
them  from  above  (see  Fig.  41).  If 
the  point  of  the  screw  is  very  sharp,  FIG.  41. 

and  the  surface  of  the  object  in  question  convex,  great 
care  is  needed  in  finding  the  maximum  diameter. 

To  determine  the  pitch  of  the  screw,  we  select  an 
object  of  known  diameter  by  means  of  a  vernier  or 
micrometer  gauge ;  we  may  determine,  for  instance, 
the  diameter  of  a  steel  bicycle  ball.  This  is  then 
fitted  as  above  (Fig.  41)  beneath  the  point  of  the 
screw,  and  the  reading  of  the  spherometer  accurately 
determined.  Subtracting  the  zero  reading,  we  have 
the  number  of  turns  made  by  the  screw  in  traversing 
the  diameter  of  the  ball.  Dividing  this  diameter  by 
this  number  of  turns,  we  have  (as  in  1"  50  II.)  the 
pitch  of  the  screw. 

Assuming  that  the  screw  has  a  uniform  pitch,  it  is 
evident  that  the  distance  traversed  by  the  point  of 
the  screw  will  always  be  given  by  the  product  of  the 
number  of  turns  and  the  pitch  of  the  screw. 

^[  53.  Determination  of  the  Span  of  a  Spherometer. 
The  span  of  a  spherometer,  or  the  average  distance  of 
its  three  feet  (d,  f,  and  g,  Fig.  38)  from  the  central 
screw  (e)  in  its  zero  position  (Fig.  40)  is  an  impor- 
tant element  in  all  calculations  relating  to  curvature 


86  THE  SPHEROMETER.  [Exp.  20. 

(see  next  experiment).  It  may  be  determined 
roughly  by  a  series  of  measurements  with  an  ordi- 
nary vernier  gauge.  If  difficulty  is  found  in  measur- 
ing directly  the  distances  in  question  from  centre  to 
centre,  an  impression  of  the  feet  and  central  screw 
may  be  taken  on  paper,  and  the  distances  thus  indi- 
rectly determined.1  For  this  purpose  the  student 
will  doubtless  prefer  to  use  a  glass  scale,  if  one  can 
be  obtained,  graduated  in  millimetres  and  tenths.  In 
such  a  scale  the  rulings  should  be  placed 
next  the  paper,  and  examined  with  a  mag- 
nifying glass. 

If  the  feet  are  blunt  (as  a  and  b  in  Fig. 
42),  the  point  of  contact  will  be  uncertain. 
FIG.  42.      jn  suc\l  a  case  the  feet  should  be  sharp- 
ened, and  the  zero  reading  re-determined. 

1  54.  Testing  a  Spherometer.  We  have  seen  that  a 
spherometer  may  be  fitted  to  a  plane  surface  (IT  51) ; 
in  the  same  way  it  may  be  adjusted  to  a  curved  sur- 
face. To  bring  this  about,  the  central  screw  must  be 
driven  forward,  if  the  surface  is  concave,  or  turned 
backward  if  the  surface  is  convex.  The  distance 
through  which  it  must  be  moved  obviously  depends 
upon  the  curvature  of  the  surface  in  question.  The 
spherometer  can  therefore  be  used  to  determine  the 
curvature  of  surfaces.  There  are,  however,  various 
sources  of  error  in  the  use  of  a  spherometer,  and  to 


1  Some  authorities  prefer  not  to  measure  directly  the  distances  (ed, 

ff,  eg.  Fig.  38)  of  the  three  feet  from  the  central  screw,  but  to  calcu- 
late the  span  by  multiplying  the  average  of  the  three  distances  (df, 

fg,  gd)  between  the  feet  by  the  square  root  of  one  third,  or  0.57735. 


H  54.]  TESTING  A  SPHEKOMETER.  87 

detect  these,  the  instrument  is  first  of  all  adjusted  to 
a  surface  of  known  curvature,  as  for  instance  that  of 
the  sphere  used  in  Experiments  8  and  9, 
(see  Fig.  43),  or  if  that  is  not  large 
enough,  to  some  other  sphere  of  known 
diameter.  The  central  screw  is  set  as  in 
f  51,  so  that  rocking  is  barely  percep- 
tible, and  the  reading  of  the  instrument 
is  determined  with  the  same  degree  of 
precision  as  before.  At  least  ten  set-  FlG-  4a 
tings  should  be  made  on  different  portions  of  the 
spherical  surface.  In  reducing  the  results  we  find 
first  the  average  reading  of  the  spherometer,  then 
subtracting  the  zero  reading  we  find  the  number  of 
turns  which  the  screw  has  made,  and  hence  the  dis- 
tance in  millimetres  through  which  the  point  of  the 
screw  has  retreated  from  its  zero  position,  since  the 
pitch  of  the  screw  has  been  already  determined  in 
IT  52.  If  this  distance  is  d,  and  the  diameter  of  the 
sphere  D,  the  square  (s2)  of  the  span  of  the  spher- 
ometer may  be  calculated  by  the  formula  (see  1"  56, 

II.),— 

sa  _  Dd  —  (P. 

In  this  formula,  all  measurements  should  be  ex- 
pressed in  millimetres.  The  result  should  confirm 
that  obtained  by  squaring  the  span  actually  observed 
in  If  53.  Slight  discrepancies  may  sometimes  be 
traced  to  obliquity  or  excentricity  of  the  central 
screw,  or  to  irregularities  in  the  shape  of  the  three 
feet. 


88  THE   SPHEROMETER.  [Exp.  21. 

EXPERIMENT  XXI. 

CURVATURE    OF    SURFACES. 

If  55.  Determination  of  the  Radius  of  Curvature  of  a 
Spherical  Surface.  It  is  frequently  required  in  optics 
to  know  the  curvature  of  the  surfaces  of  a  lens ; 
for  this  curvature,  together  with  the  nature  of  the 
glass  of  which  a  lens  is  made  determines  its  power 
of  bringing  light  to  a  "focus"  (§§  103-104)  ;  and 
conversely,  if  the  curvature  and  focussing  power 
are  known,  we  may  find  what  sort  of  glass  the  lens 
is  composed  of.  This  subject  will  be  fully  treated  of 
in  Experiments  41  and  42.  It  is  necessary  at  present 
only  to  point  out  that  as  the  surfaces  of  lenses  are 
generally  ground  to  resemble  portions  cut  out  of  a 
sphere,  their  curvature  may  be  determined  in  the 
same  way  as  that  of  any  other  spherical  surface. 

The  spherometer  is  set  upon  the  lens  as  in  Figure 
44,  and  adjusted  so  that  rocking  is  barely  perceptible 
as  in  IT  51  and  1"  54.     Ten  settings 
are   thus   made   on   each    side   of 
the  lens,  the  curvatures  of  which, 
even  if  both  are  convex,  are  by  no 
means  necessarily  the  same.     Be- 
tween    successive    measurements 
the   position  of   the  spherometer 
should  be  varied  somewhat,  so  as  to  determine  as  well 
as  possible  the  average  curvature  of  each  surface. 

The  results  are  then  averaged  for  each  surface ; 
the  mean  zero  reading  subtracted  from  each,  and  the 


IT  56.] 


CURVATURE  OF  SURFACES. 


89 


distance  (c?)  between  the  point  of  the  screw  and  the 
plane  of  its  three  feet  thus  determined.  From  this, 
the  diameter,  D,  of  the  sphere  of  which  the  surface 
in  question  forms  a  part  is  calculated  by  the  formula 
(see  IF  56,  I.), 

D  =  a  +  «2  -r-  a. 

where  s2  is  the  square  of  the  span  already  calculated 
in  the  last  article. 

The  "  radius  of  curvature  "  is  found  by  halving  the 
diameter. 

*[T  56.  Theory  of  the  spherometer.  The  formulae  of 
the  last  two  articles  depend  upon  the  following  con- 
siderations :  Let  a,  Figure  45, 
be  the  point  of  the  central 
screw  of  a  spherometer,  and 
b  one  of  the  three  feet  lying 
in  the  plane  5c,  and  let  ad  be 
a  diameter  of  the  sphere  abd 
intersecting  the  plane  be  at  c; 
then  if  the  screw  is  properly 
adjusted,  acb  and  bed  will  be 
right  triangles.  Now  abd  is 
also  a  right  triangle,  being  measured  by  half  the  semi- 
circular arc  ad;  hence  the  angles  cba  and  bdc  are 
equal,  both  being  complementary  to  cbd ;  the  right 
triangles  abc  and  bdc  are  therefore  similar  and  we 
have  — 

ac  :  be  :  :  ~bc  :  cd,  whence 

cd  =  be*  -f-  ac,  and 

ad  =  ac  -j-  cd  =  ac  -f-  Id*  -r-  ac.  I. 

We  are  thus  able  to  calculate  the  diameter  of  a 
sphere  (ac?)  if  we  know  the  span  of  the  spherometer 


FIG.  45. 


90 


EXPANSION  OF  SOLIDS. 


[Exp.  22. 


,  and  the  distance,  ac,  between  the  point  of  the 
screw,  a,  and  the  plane  of  the  three  feet,  be.  We 
can  also  calculate  the  square  of  the  span  be,  by  the 
formula,  easily  derived  from  the  above, 

be*  =  ~^d  X  ac  —  oc2.  II. 


EXPERIMENT  XXII. 

EXPANSION   OF   SOLIDS. 


^[  57.  Determination  of  the  Coefficient  of  Linear  Ex- 
pansion. —  By  measuring  the  length  of  a  rod  at  two 
different  temperatures,  the  amount  of  linear  expan- 


/r-^  •  ' 

e 

T 

^                 <•^ 

' 

^                   A 

IS 

0 

FIG.  46. 

sion  due  to  heat  may  obviously  be  determined.  To 
make  the  expansion  measurable,  a  long  rod  must  be 
employed  ;  and  even  then  delicate  instruments  are 
needed  to  measure  the  expansion  accurately.  A  mi- 
crometer gauge,  especially  constructed  for  this  pur- 
pose, is  represented  in  Fig.  46.  It  consists  of  a 
rectangular  wooden  frame,  bcon,  capable  of  admitting 
a  metallic  rod,  gi,  1  metre  long,  between  the  fixed 
point  fg  and  the  point  of  the  micrometer  screw,  i.j. 


If  57.]  LINEAR   EXPANSION.  91 

The  rod  is  surrounded  with  a  tube,  also  1  metre  long, 
held  in  place  by  the  supports,  k  and  m.  The  tube  is 
closed  at  both  ends  with  corks,  thinner  near  the 
middle  than  at  the  edges,  and  serving  to  keep  the 
rod  in  position. 

A  setting  of  the  micrometer  is  first  made  with 
the  rod  in  position,  and  the  reading  determined  (see 
Tf  50,  II.)  ;  the  temperature  of  the  rod  is  then  found 
by  means  of  a  thermometer,  A,  passing  through  a 
cork,  e,  in  the  side  of  the  tube.  To  determine  the 
pitch  of  the  micrometer,1  it  is  turned  backward  (as 
in  ^[  52)  until  an  object  of  known  diameter  fits  be- 
tween it  and  the  end  of  the  rod.  A  new  reading  is 
then  made,  and  the  pitch  of  the  screw  is  calculated 
as  in  the  case  of  an  ordinary  micrometer  gauge 

a  50,  no. 

The  screw  of  the  micrometer  is  now  withdrawn,  to 
allow  room  for  the  expansion  of  the  rod,  and  steam 
from  a  generator  (a)  is  passed  through  the  tube 
from  the  inlet  (c?)  to  the  outlet  (Z).  As  soon  as 
a  steady  current  of  steam  appears  at  the  outlet,  a 
new  setting  of  the  micrometer  is  made. 

Subtracting  from  the  last  reading  of  the  microme- 
ter the  original  reading,  we  find  the  number  of  turns 
made  by  the  screw.  From  this,  knowing  the  pitch  of 
the  screw  (^[  52),  we  find  the  expansion  of  the  rod 
in  mm.  Subtracting  the  original  temperature  (let  us 
say  20°)  from  the  final  temperature  (100°,  nearty, 
— see,  however,  Table  14)  we  find  the  rise  of  temper- 

1  By  using  the  same  micrometer  as  in  If  52,  a  determination  of 
pitch  will  be  rendered  unnecessary. 


92  EXPANSION  OF  SOLIDS.  [Exp.  22. 

ature  which  has  caused  this  expansion.  To  find  the 
expansion  of  1  mm.,  we  divide  the  total  expansion  by 
the  length  of  the  rod  in  mm.  (1,000  mm.)  ;  and  we 
divide  the  quotient  by  the  rise  of  temperature  in 
degrees  (80°  in  this  instance)  to  find  the  expan- 
sion in  mm.  of  1  mm.1  for  1°.  The  result  is  called 
the  coefficient  of  linear  expansion  of  the  material 
of  which  the  rod  is  composed  (§  83). 

^[  58.  Errors  in  the  Determination  of  Linear  Ex- 
pansion. —  In  determining  the  temperature  of  a 
metallic  rod  by  a  thermometer  beside  it,  a  consider- 
able error  is  likely  to  arise  unless  the  temperature  of 
the  surrounding  air  is  constant,  and  the  observation 
prolonged.  Air  is,  as  we  shall  see  (Experiment  31), 
a  comparatively  poor  conductor  of  heat.  To  attain 
greater  accuracy  in  this  experiment,  the  tube  may 
be  filled  with  water,  as  it  is  found  that  an  equilibrium 
of  temperature  is  reached  much  more  quickly  with 
water  than  with  air  (see  ^[  65,  (6)  ).  A  still  more 
accurate  method  is  to  replace  the  tube  by  a  trough 
packed  with  melting  ice  or  snow.  The  mixture 
should  be  stirred  vigorously  for  a  few  minutes,  so 
that  the  rod  may  acquire  a  nearly  uniform  tempera- 
ture, not  far  from  0°.  If  this  method  is  followed  an 
observation  of  the  thermometer  will  be  unnecessary. 

For  rough  purposes,  the  temperature  of  the  steam 
which  fills  the  tube  in  the  second  part  of  the  experi- 
ment may  be  assumed  to  be  100°  ;  but  this  tempera- 

1  The  student  should  note  that  the  expansion  of  1  mm.  in  mm.  is 
numerically  the  same  as  that  of  1  cm.  in  cm.  The  result  does  not 
therefore  need  to  be  reduced  to  the  C.  G.  S.  System. 


1[  58.]  LINEAR  EXPANSION.  93 

ture  really  depends  more  or  less  upon  the  barometric 
pressure.  The  thermometer  cannot  be  depended 
upon  to  give  this  temperature  correctly,  particularly 
if  the  bulb  only  is  surrounded  by  steam.  When  ac- 
curacy is  desired,  an  observation  of  the  barometer 
must  be  made  (see  IT  13).  The  true  temperature  of 
the  steam  may  then  be  found  by  Table  14,  as  will  be 
explained  in  Experiment  25. 

It  is  obviously  impossible  for  the  whole  rod,  gi 
(Fig.  46),  to  be  in  contact  with  the  steam  or  ice 
surrounding  it ;  for  even  when  the  corks  are  hol- 
lowed out,  as  shown  in  the  figure,  so  as  to  leave 
nearly  the  whole  surface  of  the  rod  uncovered,  there 
must  still  be  a  small  portion  at  each  end  which  the 
steam  or  ice  can  never  reach.  The  expansion  of  the 
rod  will  not  therefore  be  as  great  as  it  should  be. 

On  the  other  hand,  the  points  fg  and  ij\  being 
heated  by  contact  with  the  rod,  will  expand  some- 
what, and  thus  make  the  expansion  of  the  rod  ap- 
pear to  be  greater  than  it  really  is.  To  diminish  the 
conduction  of  heat,  the  teeth  may  be  protected  by 
the  use  of  insulating  material,  or  by  simply  pointing 
them.  In  all  cases  contact  should  be  maintained 
only  as  long  as  may  be  necessary  to  make  a  reading 
of  the  micrometer.  There  is  always  more  or  less  un- 
certainty as  to  temperature  when  a  hot  and  a  cold 
body  are  in  contact.  To.  eliminate  errors  arising  from 
this  source,  it  would  suffice  to  construct  a  new  ap- 
paratus, which  should  be  as  short  as  possible,  but 
otherwise  similar  to  the  first,  and  to  calculate  the 
results  from  the  difference  of  expansion  in  the  two 


94  EXPANSION  OF  LIQUIDS.  [Exp.  23. 

cases,  according  to  the  general  method  suggested  in 
§32. 

There  is,  however,  no  way  to  allow  for  the  expan- 
sion of  the  sides  of  the  gauge,  caused  by  the  warmth 
of  the  steam  jacket.  We  meet  here,  in  fact,  one  of 
the  fundamental  difficulties  in  the  accurate  measure- 
ment of  expansion,  —  namely,  changes  in  the  length 
of  the  instruments  by  which  expansion  is  measured. 
To  avoid  errors  from  this  source,  a  glass  tube  is 
sometimes  substituted  for  the  metallic  tube  repre- 
sented in  Fig.  46,  so  that  the  expansion  of  the  rod 
may  be  observed  from  a  distance.  In  the  most  ac- 
curate determinations,  the  gauge  or  standard  used  for 
comparison  is  insulated  from  all  sources  of  heat,  and 
even,  in  some  cases,  maintained  artificially  at  a  uni- 
form temperature. 

The  expansion  of  a  gauge  constructed,  like  that 
shown  in  Fig.  46,  principally  of  wood  (see  Table  8,  6), 
and  with  sufficient  space  for  the  circulation  of  air, 
will  be  found  in  practice  to  be  very  slight ;  but,  in 
the  absence  of  special  precautions,  the  student  should 
not  expect  his  results  to  contain  more  than  three 
significant  figures  (§  55). 


EXPERIMENT  XXIII. 

EXPANSION   OF   LIQUIDS,    I. 

^[  59.  Determination  of  the  Coefficient  of  Expansion 
of  a  Liquid  by  the  Method  of  Balancing  Columns.  —  A 
convenient  form  of  apparatus  for  this  experiment 


IT  59.] 


BALANCING   COLUMNS. 


95 


(see  Fig.  47)  consists  of  two  vertical  metallic  tubes, 
ch  and^)',  about  one  metre  long,  with  horizontal  elbows 
(cd,  ef,  hi,  and  ify  at  each 
end.  The  lower  elbows  are 
connected  together  with  a 
rubber  tube  (z),  while  each  of 
the  upper  elbows  is  joined  to 
one  end  of  a  differential  gauge 
(aJ)  by  one  of  the  rubber 
couplings  (d  and  e).  Each 
of  the  tubes  ch  and/)'  is  sur- 
rounded with  a  larger  tube,  or 
'•'•jacket"  which  can  be  filled 
either  with  melting  ice,  with 
water,  or  with  steam.  The 
spouts  g  and  k  are  to  be 
used  either  as  inlets  or  as  out-  3 
lets,  as  the  experiment  may 
require. 

The  liquid  whose  expansion  is  to  be  investigated 
is  first  freed  from  any  air  which  may  be  held  in  solu- 
tion, by  boiling  it,  then  poured  steadily  through  a 
funnel  into  the  tube  a  until,  after  completing  the 
circuit  (adchijfeb),  it  issues  in  a  continuous  stream 
from  b.  The  whole  apparatus  is  now  inclined  first  to 
the  right  and  then  to  the  left,  so  that  any  bubbles  of 
air  which  may  be  lodged  in  the  horizontal  tubes  may 
have  an  opportunity  to  escape.  A  little  liquid  is 
next  poured  out,  until  the  column  stands  at  the 
level  b.  This  level  should  be  the  same,  at  first,  on 
both  sides  of  the  gauge. 


FIG.  47. 


96  EXPANSION  OF  LIQUIDS.  [Exp.  23. 

Steam  is  then  admitted  to  the  jacket  eg  through 
the  spont  g  ;  and  the  jacket  fk  is  filled  with  water 
from  a  faucet  by  a  tube  connected  to  the  spout  k. 
The  temperature  of  the  water  is  observed  after  it 
reaches  the  top  of  the  tube,  /.  The  height  of  the 
liquid  in  each  side  of  the  gauge  (a  and  6)  is  measured 
as  soon  as  it  becomes  stationary,  by  means  of  a  milli- 
metre scale,  as  in  Experiment  16.  (See  ^f  42.)  The 
vertical  length  of  the  tube  (ch}  is  finally  measured  be- 
tween the  elbows  (cd  and  hi),  from  centre  to  centre, 
as  close  as  possible  to  the  jacket.  This  measurement 
should  (strictly)  be  made  while  the  tube  is  still 
heated  by  steam. 

When  the  apparatus  has  become  sufficiently  cool, 
the  water  is  emptied  out  of  the  jacket  fk,  which 
is,  in  its  turn,  filled  with  steam,  while  the  jacket 
eg  is  cooled  by  water  from  the  faucet.  The  tem- 
perature of  the  water  and  the  reading  of  the  gauge 
are  observed  as  before ;  in  this  case,  however, 
the  vertical  distance  fj  is  measured.  The  object 
of  interchanging  the  jackets  is  (see  §  44)  to  elimi- 
nate errors  due  to  capillarity,  or  in  fact  any  cause 
which  might  tend  constantly  to  raise  or  lower  the 
level  of  the  liquid  on  one  particular  side  of  the 
gauge. 

Instead  of  admitting  steam  to  one  of  the  jackets, 
melting  ice  may  be  employed,  or  water  at  various 
temperatures,  which  must,  of  course,  be  observed. 
The  other  jacket  is  always  maintained  at  a  tempera- 
ture not  far  from  that  of  the  room,  by  the  water  with 
which  it  is  filled. 


If  60.]  BALANCING  COLUMNS.  97 

^[60.  Precautions  in  determining  Expansion  by  the 
Method  of  Balancing  Columns.  —  It  is  evident  that 
the  temperatures  employed  in  this  experiment  must 
not  be  higher  than  the  boiling-point  nor  lower  than 
the  freezing-point  of  the  liquid  in  question,  and  that 
this  liquid  must  not  be  such  as  to  act  chemically  on 
the  tubes  which  contain  it.  Even  a  very  slight 
action  may  generate  a  quantity  of  gas  sufficient  to 
impair  the  accuracy  of  the  results.  The  air  dissolved 
in  the  liquid  must  be  completely  boiled  out  before  the 
experiment,  since  otherwise  bubbles  are  apt  to  form 
when  heat  is  applied.  The  tubes  should  be  large 
enough  to  allow  the  escape  of  any  air  which  may  be 
carried  into  them  while  they  are  being  filled  ;  but 
small  bubbles  can  sometimes  be  dislodged  only  by 
jarring  the  whole  apparatus. 

The  tubes  should  be  completely  surrounded  with 
the  steam,  water,  or  melting  ice  by  which  their  tem- 
perature is  to  be  regulated.  There  should  be  a  free 
vent  through  one  of  the  spouts  (g  or  k~)  for  the  water 
formed  by  the  melting  of  the  ice,  otherwise  the  tem- 
perature of  the  mixture  may  rise  above  0°.  If  steam 
is  admitted  through  one  of  these  spouts,  the  jacket 
should  be  partly  covered,  leaving  only  a  small  opening 
through  which  the  steam  should  escape  in  a  slow  but 
continuous  stream.  If  the  jackets  contain  water,  the 
latter  should  be  stirred  vigorously  to  secure  a  uni- 
formity of  temperature.  It  is  well  also,  in  this  case, 
to  find  the  reading  of  a  thermometer  at  different 
levels.  This  will  require  either  a  self-registering 
thermometer,  or  one  with  a  very  long  stem.  If  the 
7 


98  EXPANSION  OF  LIQUIDS.  [Exp.  23. 

temperature  is  not  uniform,  the  average  temperature 
must  be  calculated.1 

The  jackets  (eg  and  /&)  should  be  made  vertical 
by  a  plumb  line,  as  nearly  as  the  eye  can  judge,  and 
also  both  branches  of  the  gauge  (ai).  The  tubes  cd, 
ef,  and  hj  should  be  perfectly  horizontal,  in  those 
portions  at  least  which  are  affected  by  the  flow  of 
heat  to  or  from  the  jackets.  The  gauge  («6)  should 
be  maintained  at  a  uniform  temperature  (the  same 
always  as  that  of  one  of  the  jackets)  by  surrounding 
it,  if  necessary,  with  water.  The  tubes  of  which  this 
gauge  is  constructed  should  be  of  the  same  uniform 
calibre,  and  both  perfectly  clean,  otherwise  the  effects 
of  capillary  action  may  not  be  perfectly  eliminated. 
It  is  well  to  make  sure,  both  before  and  after  the  ex- 
periment, that  the  liquid  stands  at  the  same  level  on 
both  sides  of  the  gauge  when  the  temperature  in  the 
two  jackets  is  the  same. 

To  obtain  the  most  accurate  readings  of  such  a 
gauge,  a  double  sight  should  be  employed,  as  in  the 
case  of  a  standard  barometer.  The  setting  is  always 
made  so  that  the  plane  of  the  sights  may  be  tangent 
to  the  meniscus,  or  curved  surface  of  the  liquid  (see 
^[13  and  ^  42).  The  sights  may  be  provided  with 
a  vernier  reading  to  tenths  of  a  millimetre. 

^[  61.  Theory  of  Balancing  Columns  at  Unequal 
Temperatures.  —  The  difference  in  hydrostatic  pressure 
between  the  two  liquid  columns,  eA  andfj,  is  balanced 

1  The  average  temperature  will  be  indicated  sit  once  by  an  air 
thermometer  of  sufficient  length,  which  the  student  himself  may  be 
interested  to  construct.  See  Experiment  26. 


f  61.]  COEFFICIENTS  OF  EXPANSION.  99 

by  the  pressure  of  a  column  of  liquid  reaching  from 
a  to  5,  or  more  strictly,  by  the  difference  between  the 
hydrostatic  pressure  of  such  a  column  and  that  of  an 
equally  long  column  of  air.  The  latter,  being  exceed- 
ingly light,  may  be  left  out  of  the  account.  To  sim- 
plify calculations,  we  will  suppose  all  the  tubes  to 
have  a  cross-section  of  1  sq.  cm.  Then  if  d  is  the 
difference  in  cm.  between  the  two  levels  (a  and  b)  in 
the  gauge,  when  it  is  maintained  at  the  same  tem- 
perature (£)  as  the  jacket  fj  ;  and  if  I  is  the  length 
of  the  column  ch  at  a  higher  temperature,  ta;  then 
I  cu.  cm.  of  the  liquid  at  the  temperature  t.2  plus  d 
CM.  cm.  at  the  temperature  tv  balance  I  cu.  cm.  at  the 
temperature  tr  It  follows  that  I  cu.  cm.  at  t°  must 
balance  (I  —  d*)  cu.  cm.  at  tf.  Now  two  columns  of 
liquid  of  the  same  cross-section  cannot  balance  one 
another  unless  they  have  the  same  total  weight  ; 
hence  the  same  quantity  of  liquid  which  occupies 
(I  —  cf)  cu.  cm.  at  t°  must  expand  by  the  amount 
d  cu.  cm.  when  heated  to  t.2°,  since  it  then  occupies 
I  cu.  cm.  If  an  expansion  of  d  cu.  cm.  is  caused  by 
a  rise  of  (£2  —  ^)  degrees,  1°  would  cause  an  expan- 
sion in  the  average  (£2  —  ^)  times  less  than  d  cu.  cm.; 
and  since  the  expansion  of  1  cu.  cm.  would  be  (Z  —  c?) 
times  less  than  that  of  (I  —  cT)  cu.  cm.,  the  expansion 
(ef  of  1  cu.  cm.  for  1°  would  be 


This  expression  becomes  somewhat  modified  when 
the  gauge  is  at  the  higher  temperature,  t.2.    We  have, 


100  EXPANSION  OF  LIQUIDS.  [Exp.  23. 

then,  (Z  +  d)  cu.  cm.,  all  at  the  temperature  £2,  balancing 
I  cu.  cm.  at  the  temperature  ^.  The  expansion  is  as 
before,  d  cu.  cm. ;  but  the  quantity  expanding  is  no 
longer  {I —  d),  but  I  cu.  cm.  The  expansion  e"  per 
cu.  cm.  per  degree  is  therefore 

e"=^J—  II. 


We  have  assumed  so  far  that  the  tubes  have  a 
cross-section  of  1  sq.  cm. ;  but  the  principles  of  hy- 
drostatic pressure  are  independent  of  cross-section 
(see  §  63)  ;  hence  the  solutions  found  in  one  case 
may  be  applied  to  all.  The  method  of  balancing 
columns  is  the  only  one  which  enables  us  to  measure 
the  expansion  of  a  liquid  without  taking  into  account 
changes  in  the  capacity  of  the  vessel  in  which  the 
liquid  is  contained. 

The  object  of  this  method  is  to  determine  an  aver- 
age coefficient  of  expansion  between  two  tempera- 
tures rather  than  the  true  coefficient  of  expansion 
(§  83)  at  any  particular  temperature.  The  results 
may  differ  considerably  from  those  contained  in 
Table  11,  which  refers  in  nearly  all  cases  to  the 
expansion  of  liquids  from  0°  to  1°  Centigrade.  We 
consider,  moreover,  the  expansion  of  a  quantity  of 
liquid  measuring  1  cu.  cm.  at  the  lower  of  the  two 
temperatures  observed  instead  of  at  0°.  The  result 
given  by  the  formulae  of  this  section  should,  therefore, 
be  designated  as  the  relative  coefficient  of  expansion 
from  t°  to  t°,  that  is,  from  the  lower  to  the  higher 
temperature. 


II  62.]  COEFFICIENTS  OF  EXPANSION.  101 

EXPERIMENT  XXIV. 
EXPANSION  OF  LIQUIDS,   II. 

^|  62.  Determination  of  the  Coefficient  of  Expansion 
of  a  Liquid  by  means  of  a  Specific  Gravity  Bottle.  — 
The  experiment  consists  essentially  of  a  repetition 
of  Experiment  14,  with  a  given  liquid  at  two  or 
more  different  temperatures.  These  temperatures 
should  be  separated  from  one  another  as  widely  as 
possible,  in  order  that  the  densities  observed  may 
differ  by  an  amount  large  enough  to  be  accurately 
measured.  The  temperatures  themselves  must  be 
determined  with  the  greatest  care,  particularly  if 
they  are  far  above  or  far  below  the  temperature  of 
the  room ;  for  in  this  case  rapid  changes  will  take 
place  and  must  be  guarded  against. 

A  convenient  way  of  heating  a  liquid  in  a  specific 
gravity  bottle  to  a  uniform  temperature,  is  to  sur- 
round the  bottle  up  to  the  neck  with  $iot  water. 
To  prevent  evaporation,  the  bottle  should  be  closed 
temporarily  by  a  cork,  with  a  hole  made  in  it 
sufficiently  large  to  admit  freely  the  stem  of  a  ther- 
mometer, to  which  a  brass  fan  is  attached  (see  Fig. 
50,  ^[  65).  By  this  means  the  liquid  is  continually 
stirred  until  a  maximum  temperature  is  reached. 
As  soon  as  the  reading  of  the  thermometer  has  been 
observed,  the  stopper  is  inserted,  with  due  care  not  to 
enclose  bubbles  of  air  (see  ^[  32,  Fig.  19).  The  bottle 
is  then  carefully  dried,  and  weighed  at  leisure  (see 
^[  33), after  cooling  to  the  temperature  of  the  room. 


102  EXPANSION  OF  LIQUIDS.  [Exp.  24. 

The  student  is  advised  not  to  attempt  determina- 
tions of  density  below  the  temperature  of  the  room, 
on  account  of  the  obvious  difficulty  of  preventing 
the  loss,  especially  in  the  case  of  a  volatile  liquid, 
of  the  portion  which  is  forced  out  of  a  specific 
gravity  bottle  by  .its  gradual  rise  of  temperature. 
He  should,  however,  make  at  least  two  determina- 
tions of  density  above  the  temperature  of  the  room, 
with  the  liquid  already  employed  in  Experiment  14 ; 
and  he  should  repeat  rapidly  the  determination  made 
in  that  experiment  at  the  temperature  of  the  room, 
to  make  sure  that  the  result  has  not  been  seriously 
affected  by  atmospheric  changes,  or  by  variations  of 
the  density  of  the  liquid  due  to  evaporation  or  other 
causes.  Coefficients  of  expansion  are  then  calculated 
and  reduced  as  explained  in  the  next  section. 

^[  63.  Calculation  of  Coefficients  of  Expansion.  — 
Let  £j,  £2,  ts,  etc.,  be  the  temperatures  at  which  the 
densities  dn  c?2,  rfg,  etc.,  respectively,  have  been  deter- 
mined and  calculated,  essentially  as  in  ^[  38.  The 
results  are  first  represented  by  points  plotted  on  co- 
ordinate paper  (see  Fig.  48),  and  connected  by  a 
curve  drawn  with  a  bent  ruler,  essentially  as  in  §  59. 
The  necessary  forces  should  be  applied  to  the  ruler 
as  near  the  ends  as  possible,  in  order  that  the  curve 
may  be  continued  downward  as  far  as  0°.  The 
density  of  the  liquid  (d0)  at  0°  is  now  inferred  by 
means  of  this  curve  (see  §  59). 

The  specific  volumes,  v0,  vt,  v2,  v3,  etc.,  correspond- 
ing to  the  densities  d0,  dl9  c?2,  ds,  etc.,  are  now  found 
by  the  formulae  derived  from  ^f  37,  — 


IT  63.] 


COEFFICIENTS   OF   EXPANSION. 


103 


va  =  1  -i-  du  ;  vi  =     -i-l;  v2  =     -j-  c2;  v8  ==     -5-  J3, 

etc.  Evidently  a  certain  quantity  of  liquid  expands 
by  the  amount  (v9  —  vj  cu.  cm.  when  heated  from  the 
temperature  ^  to  the  temperature  tz  ;  that  is,  (£2  —  ^) 
degrees.  The  expansion  per  degree  is  therefore 
(vz  —  Vl)  -f-  (£2  —  ^).  Since  the  quantity  of  liquid 


&-Q011-  0012  •  0013-OOtV  00(5 


FJG.  48. 


FIG. 


thus  expanding  occupies  v0  cu.cm.  at  0°,  the  ex- 
pansion (e)  of  a  quantity  occupying  1  cu.  cm.  at  that 
temperature  would  be  one  v0ih  as  large,  or 


The  coefficient  e  which  determines  the  expansion 
of  a  quantity  of  liquid  occupying  the  unit  of  volume 
at  the  standard  temperature  (0°)  is  a  true  as  distin- 
guished from  a  relative  coefficient  of  expansion  (see 
^  61)  ;  it  expresses,  however,  the  average  expansion 
between  the  two  temperatures  t^  and  £2.  We  find  in 
the  same  way  the  average  coefficient  of  expansion  from 
£2  to  £3  by  substituting,  in  the  formula  above,  £2,  £3,  t>2, 
and  v3,  for  £,,  £2,  v^  and  v2,  respectively.  Each  result 
may  be  represented  on  co-ordinate  paper  by  a  cross, 
at  the  right  of  a  point  half-way  between  the  two 
temperatures  in  question,  and  under  the  correspond- 


104  EXPANSION  OF  LIQUIDS.  [Ex  p.  25. 

ing  coefficient  of  expansion  (see  Fig.  49).  A  line 
drawn  through  these  points  represents  approximately 
the  coefficient  of  expansion  at  any  given  tempera- 
ture. It  is  clear,  however,  that  with  only  two  de- 
terminations of  the  coefficient  of  expansion,  we  can- 
not tell  even  whether  this  line  should  be  straight  or 
curved. 

EXPERIMENT  XXV. 

THE   MERCURIAL   THERMOMETER. 

^[  64.  Preservation  of  a  Mercurial  Thermometer.  — 
It  would  seem  hardly  necessary  to  point  out  that  a 
mercurial  thermometer  is  an  exceedingly  fragile  in- 
strument ;  but  in  the  processes  of  manipulation  about 
to  be  described,  it  is  frequently  required  that  a  ther- 
mometer should  be  subjected  to  forces  very  near  the 
limit  of  its  strength,  and  which,  even  in  skilled 
hands,  may  break  it.  The  student  is  therefore  ad- 
vised to  experiment  with  thin  tubes  or  strips  of  win- 
dow-glass, before  attempting  the  calibration  of  a 
thermometer ;  and  to  examine  the  almost  micro- 
scopic thickness  of  the  glass  constituting  the  bulb, 
before  subjecting  it  to  any  considerable  pressure. 
In  respect  to  its  resistance  to  a  blow  endwise,  the 
bulb  of  a  thermometer  may  perhaps  be  compared 
to  the  point  of  a  lead-pencil  when  moderately  sharp. 
In  attempting  to  move  the  mercury  in  the  thermome- 
ter by  centrifugal  force,  the  student  should  limit 
himself  to  such  velocities  as  he  might  give  to  a 
palm-leaf  fan.  More  thermometers  are  broken  by 


t  65.]      THE  MERCURIAL  THERMOMETER.      105 

suddenly  arresting  than  by  suddenly  creating  the 
necessary  velocity.  If  a  glass  thermometer  be  tem- 
porarily mounted  on  a  wooden  support,  like  an  or- 
dinary house  thermometer,  it  may  be  much  more 
roughly  treated  with  the  same  safety. 

The  full  heat  of  a  flame  should  never  be  applied 
immediately  to  any  glass  instrument,  since  fracture 
will  almost  inevitably  result.  By  giving  to  a  flame 
a  waving  motion,,  heat  may  be  applied  as  slowly  as 
may  be  desired.  As  soon  as  the  glass  acquires  a  dull- 
red  heat  the  danger  of  fracture  is  past.  There  will, 
however,  be  no  occasion  for  so  high  a  temperature  in 
the  case  of  a  thermometer.  The  student  is  particu- 
larly cautioned  against  plunging  a  cold  thermometer 
into  hot  mercury,1  or  a  hot  thermometer  into  any 
cold  liquid  whatsoever. 

In  applying  heat  to  the  bulb  of  a  thermometer, 
care  must  be  taken  not  to  drive  out  more  mercury 
than  there  is  room  for  in  the  expansion  chamber 
at  the  top  of  the  instrument.  The  temperature  of 
the  mercury  should  not  be  raised  above  its  boiling- 
point  2  (350°  C.)  in  any  part  of  the  thermometer ; 
for  the  pressure  of  the  vapor,  being  transmitted  to 
the  bulb,  will  be  likely  to  cause  an  explosion. 

^[  65.  Precautions  in  the  Use  of  a  Mercurial  Ther- 
mometer. —  (1)  TEMPER.  —  In  addition  to  the  dan- 

1  The  thermometer  should  be  placed  in  the  mercury  while  cold, 
and  gradually  heated  with  the  mercury.     On  account  of  its  rapid 
conduction  of  heat,  mercury  is  more  likely  to  cause  fracture  than 
other  liquids. 

2  Special  thermometers  are  now  constructed  so  as  to  read  safely  a.s 
high  as  the  boiling  point  of  sulphur  (440°  C.). 


106  EXPANSION  OF  LIQUIDS.  [Exp.  25. 

ger  of  fracture,  the  accuracy  of  a  thermometer  may 
be  greatly  impaired  by  any  wide  change  of  tem- 
perature, especially  if  the  change  be  sudden.  After 
a  thermometer  is  freshly  made,  there  is  found  to  be 
a  gradual  contraction  of  the  bulb,  which  continues 
perceptibly  for  months  and  even  for  years.  This 
accounts  for  the  fact  that  nearly  all  old  thermome- 
ters stand  somewhat  too  high,  although  they  are  not 
supposed  to  be  graduated  until  'the  contraction  of 
the  bulb  has  ceased.  The  value  of  a  thermometer 
evidently  depends  partly  on  its  age  or  "temper." 
This  value  may  be  completely  destroyed  by  a  sudden 
change  of  temperature. 

(2)  CHANGE  OF  FIXED  POINTS.  —  In  fact,  when 
a  thermometer  is  simply  heated   to  the  temperature 
of  steam,  then  cooled  as  gradually  as  possible,  the 
readings  are  almost  always  affected  to  the  extent  of 
one   or   two   tenths   of  a  degree.     In  the  course  of 
a  month  the  thermometer  may  return  to  its  former 
reading,  but  the  change  is  gradual.     It  is  therefore 
customary   to   test  a  thermometer  —  in  ice,  for  in- 
stance —  (see  ^j  69,  II.)  after  testing  it  in  steam  (see 
^[  69,  I.),  or  in  fact  after  subjecting  it  to  any  consid- 
erable change  of  temperature. 

(3)  CONTINUITY  OF  THE  MERCURIAL  COLUMN. 
—  Errors  in  reading  a  thermometer  frequently  arise 
from  a  break  in  the  mercurial  column,  which  can  be 
guarded  against  only  by  inspection.     A  slight  jar- 
ring is  usually  sufficient  to  make  the  column  reunite  ; 
but  when  a  small  bubble  of  air  interrupts  the  col- 
umn, or  when  in  the  expansion  chamber  a  globule 


165.]  THE   MERCURIAL  THERMOMETER.  107 

becomes   separated   from   the   rest  of  the    mercury, 
special  precautions  are  necessary  (see  ^[^[  65,  67). 

(4)  TEMPERATURE  OF  THE  STEM.  —  To  make  an 
accurate  determination  of  temperature  with  a  mercu- 
rial thermometer,  it  is  necessary  that  the  mercury,  in 
the  stem  as  well  as  in  the  bulb,  should  be  raised  to  the 
temperature  in  question.     In  a  thermometer  reading 
to  — 10°  C.,  for  instance,  if  the   bulb  only  is  heated, 
the   errors,    even    if    the    thermometer    is    correctly 
graduated,  will   be  as  follows:     at  50°,  — 0°.5 ;    at 
100°,  —  2°.0  ;  at  200°,  —  7°.6 ;  at  300°,  —  17°  ;  etc. 
As  the   temperature  rises,  more  mercury  flows  into 
the  stem,  and  it  becomes  still  more  important  to  heat 
this  mercury  to  the  given  temperature  (see  ^[  84). 

(5)  UNIFORMITY  OF  TEMPERATURE.  —  In  nearly 
all  determinations  of  the  temperature  of  liquids,  it 
is  necessary  to  make  use  of  some  stirring 
apparatus,  to  secure  a  uniformity  of  tem- 
perature.   A  small  fan  of  thin  sheet  brass 

is  customarily  attached  to  the  stem  of  the 
thermometer,  just  above  the  bulb.  .  The 
stirring  is  accomplished  by  twisting  the 
stem  of  the  thermometer.  Special  de- 
vices are  necessary  when  finely  divided  substances 
are  employed,  though  the  stem  of  the  thermometer 
itself  may  (with  due  care)  occasionally  be  used, 
especially  in  mixtures,  as  of  powdered  ice  and  water, 
where  the  resistance  will  be  exceedingly  small. 

(6)  TIME  REQUIRED. — The  length  of  time  required 
to  attain  an  equilibrium  of  temperature  depends  largely 
upon  the  conductivity  of  the  surrounding  medium,  and 


108  EXPANSION  OF  LIQUIDS.  [Exp.  25. 

upon  the  degree  of  accuracy  which  is  aimed  at.  Let  us 
suppose  that  a  thermometer  is  taken  out  of  a  mixture 
of  ice  and  water,  and  placed  in  air  at  32° ;  if  at  the 
end  of  one  minute  it  rises  16°,  that  is,  half-way 
towards  its  final  temperature,  we  may  expect  it  to 
accomplish  in  another  minute  half  of  what  is  left, 
or  8°,  according  to  the  general  law  explained  in  §  89. 
The  temperatures  attained  would  thus  be  as  follows : 
in  1  m.,  16°;  in  2  m.,  24°;  in  3  m.,  28°;  in  4  m.,  30°; 
in  5  m.,  31°;  in  6  m.,  31 J,  etc.  At  the  end  of  10 
minutes  the  reading  would  differ  from  32°  by  only 
3^  of  a  degree,  a  quantity  hardly  perceptible  to  the 
eye  on  an  ordinary  thermometer.  Now,  if  the  ther- 
mometer had  been  placed  in  water  at  32°  instead  of 
in  air,  the  temperature  would  have  reached  16°  in  a 
few  seconds;  and  at  the  end  of  a  minute  it  would 
have  indicated  32°  within  a  very  small  fraction  of  a 
degree.  Again,  a  mixture  of  hot  lead  and  cold  water 
may  take  several  minutes  before  the  temperature  is 
practically  equalized. 

One  almost  always  knows,  at  least  roughly,  what 
the  final  temperature  will  be.  A  useful  rule  is  to 
observe  how  long  it  takes  the  temperature  to  reach 
a  point  half-way  between  its  original  and  its  final 
value;  then  to  allow  from  ten  to  twenty  times  as 
long  a  time  before  making  a  determination  of  the 
temperature,  according  to  the  degree  of  accuracy 
required. 

(7)  OTHER  PRECAUTIONS.  —  The  necessity  of 
shielding  a  thermometer  from  radiation  has  been 
already  alluded  to  (^f  15).  Delicate  thermometers 


1  66  J      THE  MERCURIAL  THERMOMETER.      109 

may  be  perceptibly  affected  by  mechanical,  hydro- 
static, or  even  barometric  pressure  on  the  bulb,  and 
by  mercurial  pressure  from  within.  Such  thermome- 
ters should  be  tested  both  in  a  vertical  and  in  a 
horizontal  position.  Other  special  precautions  will 
be  mentioned  as  the  necessity  for  them  arises. 

^[  66.  Selection  of  a  Mercurial  Thermometer.  —  For 
the  purpose  of  calibration,  it  is  best  to  select  a  glass 
thermometer,  graduated  on  its  own  stem  (be,  Fig. 
51),  in  degrees  at  least  1  mm.  long,  from  0°  to  100° 
centigrade,  with  a  few  divisions  above  100°  and 
below  0°.  The  bulb  (a&)  should  have  a  volume 1  of 
nearly  1  cu.  cm. ;  and  the  expansion  chamber  (c)  at  the 
top  of  the  thermometer  should  have  about  ^  of  this 


FIG.  51. 

capacity.  The  bulb  (a6)  should  for  convenience  be 
elongated  as  in  the  figure,  so  as  to  pass  freely  through 
a  hole  in  a  cork  fitted  to  the  stem  of  the  thermometer. 
The  expansion  chamber  should  be  pear-shaped  (see  c, 
Fig.  51),  since  otherwise  particles  of  mercury  are 
likely  to  lodge  there.  The  shape  and  size  of  the  tube 
must  be  such  that  mercury  may  be  made  to  flow,  with 
a  little  jarring,  from  one  end  to  the  other ;  and  the 
quality  of  the  mercury  such  that  there  is  no  tendency 
for  the  column  to  break  up  into  small  fragments. 

1  The  volume  of  a  thermometer  bulb  may  be  estimated  by  the 
quantity  of  water  displaced  in  a  small  measuring  glass  (IT  85).  A 
small  bulb  usually  implies  a  stem  of  small  calibre,  which  may  give 
rise  to  difficulty  in  calibration. 


HO  EXPANSION  OF  LIQUIDS.  [Exp.  25. 

v  ^[  67.  Manipulation  of  a  Thread  of  Mercury.  —  It 
is  frequently  required  in  the  calibration  of  a  ther- 
mometer to  separate  from  the  rest  of  the  mercury  in 
the  stem  of  the  thermometer  a  thread  or  column  of 
a  given  length,  and  to  place  it  in  a  given  part  of  the 
stem.  When  a  thread  has  been  broken  off,  it  may  be 
easily  moved  (by  sufficiently  inclining  or  swinging 
the  thermometer)  under  the  influence  of  its  own 
weight  or  inertia.  For  slight  motions,  jarring  is 
often  efficient.  The  place  where  the  thread  breaks 
off  is  generally  determined  by  a  microscopic  bubble 
of  air.  To  find  the  location  of  this  bubble,  the  ther- 
mometer is  inverted.  If  a  thread  of  mercury  sepa- 
rates at  once  from  the  rest,  the  position  of  the  bubble 
is  evident ;  if  the  mercury  runs  in  an  unbroken  col- 
umn into  the  expansion  chamber,  a  small  quantity  of 
air  will  probably  be  found  in  the  bulb  ;  and  if  the 
mercury  flows  easily  back  again,  there  is  probably  a 
little  air  in  the  expansion  chamber. 

The  (nearly)  empty  space  in  the  bulb  caused  by 
the  flow  of  mercury  into  the  expansion  chamber  has 
in  any  case  the  appearance  of  a  bubble,  which  may 
be  made  to  rise  into  the  neck  (5,  Fig.  51)  by  sud- 
denly turning  the  thermometer  into  an  upright  posi- 
tion. If  it  really  contains  air,  it  may  be  worked  up 
into  the  stem  by  jarring  the  thermometer,  especially 
before  all  the  mercury  has  had  time  to  flow  back  from 
the  expansion  chamber.  If  the  experiment  has  been 
successful,  a  thread  of  mercury  may  now  be  broken 
off  by  inverting  the  thermometer,  and  tapping  it 
gently  on  the  table. 


IF  67.]  THE  MERCURIAL  THERMOMETER.  Ill 

In  the  absence  of  air  in  the  bulb  or  in  the  stem,  it 
remains  only  to  make  use  of  air  in  the  expansion 
chamber.  As  much  mercury  as  possible  is  first  made 
to  flow  into  the  expansion  chamber,  and  detached 
from  the  rest  by  jarring  the  thermometer  while  in  a 
horizontal  position.  Then  the  rest  of  the  mercury 
is  returned  to  the  bulb.  If  there  is  any  air  in  the 
expansion  chamber,  a  part  of  it  will  now  flow  into 
the  bulb  ;  and  when  the  globule  of  mercury  is  once 
more  returned  to  the  bulb  by  centrifugal  force 
(see  Tf  64),  a  thread  of  mercury  can  probably  be 
separated. 

The  presence  of  a  bubble  of  air1  in  the  neck  of  the 
bulb  (6)  greatly  facilitates  the  adjustment  of  the 
length  of  the  thread  of  mercury  which  will  break 
off  when  the  thermometer  is  inverted.  If  the  bulb 
is  slowly  heated  or  cooled  by  a  certain  number  of 
degrees,  the  mercury  will  usually  flow  by  the  bubble 
without  dislodging  it,  thus  lengthening  or  shortening 
the  thread  by  that  same  number  of  degrees.  The 
surest  way,  however,  of  shortening  a  thread  of  mer- 
cury by  a  few  degrees  is  to  hold  the  thermometer 
upright  and  jar  it  slightly  (see  ^[  64),  so  that  the 
bubble  may  rise  farther  and  farther  into  the  stem. 
If  at  the  same  time  the  bulb  is  gradually  cooled,  one 
may  be  perfectly  sure  of  shortening  the  thread  to 
any  extent.  There  is  no  certain  method  of  increasing 
the  length  of  a  thread  of  mercury,  except  by  trans- 
ferring it  to  the  expansion  chamber,  and  adding  to 

1  Few,  if  any,  thermometers  will  be  found  to  be  entirely  free 
from  air. 


112  EXPANSION  OF  LIQUIDS.  [Exp.  25. 

the  globule  thus  formed  more  or  less  mercury  from 
the  stem.  The  globule  is  then  detached  and  forced 
backward  into  the  stem,  as  has  been  previously  de- 
scribed. To  prevent  it  from  all.  returning  to  the  bulb, 
the  latter  should  be  warmed  somewhat.  The  thread 
will  now,  probably,  be  much  too  long ;  but  may,  as 
we  have  seen,  be  shortened  at  pleasure. 

Certain  difficulties  which  are  occasionally  met  in 
these  manipulations  may  be  avoided  by  the  cautious 
application  of  heat  (*f[  64).  It  is  sometimes  impos- 
sible to  force  mercury  from  the  expansion  chamber 
into  the  stem  either  through  its  weight  or  through 
its  inertia,  especially  when  through  accident  the  ex- 
pansion chamber  has  been  allowed  to  become  com- 
pletely full.  Heat  should  then  be  applied  to  the  top 
of  the  expansion  chamber  until  the  mercury  is  driven 
out  by  the  pressure  of  its  own  vapor.  When  a 
thread  of  mercury  can  be  broken  off  in  no  other  way, 
heat  may  be  applied  to  the  stem  of  the  thermometer 
at  the  point  where  a  separation  is  desired.  When  the 
mercury  refuses  to  leave  the  bulb,  the  flow  may  be 
started  by  slightly  warming  it ;  in  fact,  any  desired 
quantity  of  mercury  may  be  forced  into  the  expan- 
sion chamber  in  this  way  (see,  however,  *fl"  65,  (1)). 

When  the  calibration  of  a  thermometer  has  been 
finished,  as  will  be  explained  in  the  next  section, 
it  is  well  to  remove  the  bubble  of  air  from  the  mer- 
cury. This  is  done  either  by  cooling  the  bulb  in 
a  freezing  mixture  (as,  for  instance,  ice  and  salt) 
until  no  mercury  remains  in  the  stem  ;  or  if  this  is 
impossible,  by  heating  the  bulb  until  the  air  is  driven 


: 


68.]     THE  MERCURIAL  THERMOMETER.      113 


ito  the  expansion  chamber.  In  either  case  a  slight 
jarring  should  free  the  bubble  from  the  mercury. 
If  the  bubble  is  too  small  to  respond  to  this  treat- 
ment, it  will  hardly  affect  the  accuracy  of  results, 
unless  it  actually  causes  a  break  in  the  mercurial 
column  (see  ^[  65,  (3)  ). 

^[68.  Calibration  of  a  Mercurial  Thermometer.  — 
A  thread  of  mercury,  about  50°  in  length,1  is  placed 
so  as  to  reach  first  from  0°  upwards,  then  from  100° 
downwards.  The  reading  of  the  end  near  50°  is 
taken  to  a  tenth  of  a  degree  in  both  cases,  as  will  be 
explained  below.  This  enables  us  to  detect  any  dif- 
ference in  calibre  between  the  upper  and  lower  parts 
of  the  thermometer.  Next,  a  thread  about  25°  long 
is  made  to  reach  first  from  0°,  then  from  50°  upwards, 
then  also  from  50°  and  from  100°  downwards,  with 
exact  readings  of  the  end  near  25°  or  75°,  as  the 
case  may  be.  These  will  enable  us  to  compare  the 
different  quarters  of  the  tube  from  0°  to  100°.  It 
is  not  necessary,  for  most  purposes,  to  carry  the  pro- 
cess of  calibration  any  further. 

To  avoid  parallax  (§  25)  the  eye  may  be  held  so 
that  the  divisions  of  the  scale  seem  to  coincide  with 
their  own  reflections  in  the  thread  of*  mercury.  One 
end  of  the  thread  is  always  placed  so  as  to  coincide  ex- 
actly with  a  given  division  line  of  the  scale  (0°,  50°, 
or  100°),  so  that  any  error  in  the  estimation  of 
tenths  of  degrees  will  be  confined  to  the  reading  of 
the  other  end.  To  reduce  this  error  to  a  minimum, 

1  A  thread  from  49°  to  51°  will  answer.  In  cases  presenting 
special  difficulty,  a  greater  latitude  may  be  allowed. 


114  EXPANSION  OF  LIQUIDS.  [Exp.  25. 

the  student  is  advised  to  study  or  to  construct  for 
himself  diagrams  like  the  following  (Fig.  52),  show- 
ing the  appearance  of  a  mercurial  column  when 
dividing  the  space  between  two  lines  into  a  given 
number  of  tenths,  and  to  identify  the  reading  in 
each  case  with  the  diagram  which  it  most  resembles. 

Before  calculating  a  table  of  corrections  (see  *f[  70) 
from  the  results  of  calibration,  it  is  necessary  to  de- 
termine two  "  fixed  points  "  on  the  scale  of  the  ther- 
mometer, as  will  be  explained  in  the  next  section. 


FIG.  52. 

If  69.  Determination  of  the  Fixed  Points  of  a  Thei> 
mometer.1 — I.  The  mercurial  thermometer  is  placed 
in  a  steam  generator  (Fig.  53)  so  that  the  bulb  and 
nearly  the  whole  of  the  stem  may  be  surrounded 
with  steam.  Only  the  divisions  above  99°  project 
above  the  cork  (a)  by  which  the  thermometer  is  held 
in  place.  When  the  greatest  accuracy  is  desired,  the 
sides  of  the  generator  are  made  double,  as  in  Fig.  54. 
By  this  means  the  inner  coating,  being  surrounded 
on  both  sides  with  steam,  will  have  a  temperature  of 
100°  nearly,  and  there  will  be  no  radiation  of  heat 
between  it  and  the  thermometer,  since  radiation  de- 
pends upon  a  difference  of  temperature  (§  89).  It  is 

1  The  student  who  is  interested  in  the  changes  produced  in  a  ther- 
mometer by  the  application  of  heat  will  do  well  to  observe  the 
freezing-point  before  as  well  as  after  the  boiling-point. 


169.] 


THE  MERCURIAL  THERMOMETER. 


115 


important  also  to  construct  a  shield  of  some  sort  so 
that  the  boiling  water  in  the  bottom  of  the  apparatus 
may  not  be  spattered  upon  the  bulb  of  the  thermom- 
eter. Such  a  shield  is  moreover  useful  in  prevent- 
ing the  thermometer  from  dipping  into  the  water. 
It  must  be  borne  in  mind  that  the  temperature  of 
boiling  water  is  very  uncertain,  being  sometimes 


FIG.  53. 


FIG.  54. 


several  degrees  above  the  true  boiling  temperature, 
even  when  the  water  is  perfectly  pure,  owing  to  the 
adhesion  of  the  liquid  to  the  sides  of  the  vessel  con- 
taining it.  On  the  other  hand,  the  temperature  at 
which  steam  condenses  depends  only  upon  the  pres- 
sure to  which  it  is  subjected. 


116  EXPANSION  OF  LIQUIDS.  [Exr.  25. 

It  is  possible,  with  an  apparatus  like  that  shown 
in  Fig.  53,  particularly  if  the  spout  (6)  be  small,  to 
generate  steam  so  rapidly  that  the  pressure  may  be 
perceptibly  greater  within  the  generator  than  it  is 
outside.  Care  must  be  taken  to  check  the  supply  of 
heat  until  the  feeblest  possible  current  of  steam 
issues  continuously  from  the  spout.  The  atmospheric 
pressure  is  then  to  be  observed  by  means  of  a  barom- 
eter (  [4]  Fig.  53),  and  the  reading  of  the  thermom- 
eter determined  within  a  tenth  of  a  degree  (see  ^[  68, 
Fig.  52).  If  the  barometer  happens  to  stand  at 
76  cm.,  this  reading  is  called  the  "boiling-point"  of 
the  thermometer,  otherwise  a  correction  must  be 
applied,  as  will  be  explained  in  the  next  section. 

II.  The  thermometer  is  now  allowed  to  cool  as 
slowly  as  possible  to  the  temperature  of  the  room,  so 
as  not  to  destroy  its  "  temper"  (^[  65,  (1) ), 
then  surrounded  in  a  beaker  with  a  mixt- 
ure of  water  and  finely-powdered  ice  (Fig. 
55),  well  stirred  and  covering  the  scale 
within  one  or  two  divisions  of  the  zero 
mark.  The  melting-point  of  ice  is  not 

•n  ff  O      XT 

perceptibly  affected  by  barometric  or  ordi- 
nary mechanical  pressure.  The  ice  must  be  pure 
and  clean.  The  bulb  of  the  thermometer  must  not 
be  jammed  by  the  ice  (^j  65,  (7) ).  The  reading  is 
to  be  accurately  observed  (^[  68).  This  reading  is 
called  the  "  freezing-point "  of  the  thermometer. 

The  boiling  and  freezing  points  are  called  the  two 
"fixed  points"  of  a  thermometer,  and  from  them, 
with  the  results  of  calibration,  a  complete  table  of 


t  70.J  THE   MERCURIAL  THERMOMETER.  117 

corrections  should  be  calculated,  as  will  be  explained 
in  the  next  section. 

^[  70.  Calculation  of  a  Table  of  Corrections  for  a 
Thermometer.  —  The  correction  of  a  thermometer  at 
0°  is  found  at  once  by  reversing  the  sign  of  the  read- 
ing in  melting  ice  (see  ^  69,  II.,  also  ^  41).  If,  for 
instance,  the  reading  in  melting  ice  is  -fO°.9,  the 
correction  at  0°  is  —  0°.9.  The  correction  at  100°  is 
found  by  subtracting  (algebraically)  the  actual  read- 
ing in  steam  from  the  true  temperature  of  steam  cor- 
responding to  the  barometric  pressure  observed. 
(See  Table  14.)  Thus  if  the  thermometer  reads 
99°. 0  when  the  barometer  stands  at  72  cm.,  since  the 
true  temperature  of  steam  at  this  pressure  is  98°.5, 
the  thermometer  stands  too  high  by  0°.5,  and  the  cor- 
rection is  —  0°.5.  It  is  obvious  that  under  the  nor- 
mal pressure  (76  cm.)  the  thermometer  would  indicate 
100°. 5  instead  of  100°.0^  hence  the  standard  boiling- 
point  is  100°.5  on  this  thermometer.  We  find  the 
standard  boiling-point  in  general  by  adding  (numeri- 
cally) to  100°.0  the  correction  (at  100°)  if  the  ther- 
mometer is  found  to  stand  too  high,  or  subtracting 
the  same  if  the  thermometer  stands  too  low. 

Let  us  now  suppose  that  in  the  calibration  of  the 
thermometer  a  given  thread  of  mercury  reached  from 
0°  to  49°. 5  ;  if  the  bottom  of  this  thread  had  been 
placed  at  the  observed  freezing-point  (-}-00.9)  instead 
of  at  the  mark  0°,  it  would  evidently  have  reached 
farther  up  the  tube.  Since  the  length  of  the  thread 
can  hardly  vary  by  a  perceptible  amount  when  it  is 
moved  less  than  one  degree,  even  in  a  tube  with 


118  EXPANSION  OF  LIQUIDS.  [Exp.  25. 

considerable  variations  of  calibre,  we  may  assume 
that  the  thread  would  reach  a  point  just  nine  tenths 
of  a  degree  higher  than  before  ;  in  other  words,  it 
would  reach  from  0°.9  to  50°. 4.  In  the  same  way,  if 
the  thread  is  found  to  reach  from  100°  to  50°.7,  we 
infer  that  it  would  have  reached  from  the  standard 
boiling-point  (found  by  observation  to  be  at  100°.5) 
to  a  point  five  tenths  of  a  degree  above  50°. 7,  or 
51°. 2.  Between  50°.4,  and  51°. 2  we  find  a  half-way 
point1  on  the  thermometer,  namely  50°. 8.  If  the 
thread  of  mercury  had  been  four  tenths  of  a  degree 
longer  it  would  have  reached  to  this  half-way  point, 
either  from  the  freezing-point  or  from  the  boiling- 
point.  We  infer  that  the  volume  of  the  tube  in- 
cluded between  the  boiling  and  freezing  points  is 
exactly  halved  at  50°. 8.  Now,  by  definition,  the  tem- 
perature at  which  the  mercury  reaches  this  point  is 
50°.0,  according  to  a  perfect  mercurial  thermometer; 
hence  the  correction  for  the  thermometer  at  50°  is 
-0°.8. 

In  the  same  way  we  find  the  correction  of  the 
thermometer  at  25°,  then  at  75°,  by  considering  how 
far  the  shorter  thread  (25°  long)  would  have  reached 
if  one  end  had  been  placed  at  -}-00.9  instead  of  0°, 
at  50°.8  instead  of  50°,  or  at  100°.5  instead  of  100°. 
We  thus  find  two  points  near  25°,  and  half-way  be- 
tween them  a  third  point,  showing  where  the  ther- 
mometer would  stand  at  a  temperature  of  25°, 

1  This  point  is  sometimes  called  the  "  middle  point "  of  a  ther- 
mometer ;  but  some  authorities  mean  by  the  "  middle  point "  one 
half-way  between  the  divisions  numbered  0°  and  100°  respectively. 


U  71.]  THE  AIR  THERMOMETER.  119 

according  to  a  perfect  mercurial  thermometer  ;  we  find 
also  the  indication  of  the  thermometer  for  a  temper- 
ature of  75°  ;  and  hence  also  the  corrections  at  25° 
and  75°. 

The  corrections  at  5°,  10°,  15°,  etc.,  up  to  100°  are 
finally  calculated  by  interpolation.  Thus  if  the  cor- 
rection at  25°  is  found  to  be  —  0°.8,  and  at  75°,  —  0°.7, 
we  should  find  the  following  table  :  — 

TABLE  OF  CORRECTIONS. 

0°  — 0°.9  25°  — 0°.8  50°  — 0°.8  75°  — 0°.7 

50  — o°.9  30°  — 0°.8  55°  — 0°.8  80°  — 0°.7 

IQO  — 0°.9  35°  —  0°.8  60°  — 0°.8  85° 

150  _oo.8  40°  — 0°.8  65°  — 0°.7  90° 

20°  — 0°.8  45°  — 0°.8  70°  — 0°.7  95° 

250  o°.8  60°  — 0°.8  75°  — 0°.7  100°  — 0°.5 


EXPERIMENT  XXVI. 

THE   AIR   THERMOMETER,   I. 

^[71.     Calibration    of     an     Air    Thermometer.  —  A 

simple  form  of  air  thermometer  consists  of  a  glass 

tube  (ac,  Fig.  56)   about  40  cm.  long,  and  2  mm.  in 

diameter,  closed  at  one  end  (a).     The  tube  has  an 

a . /  <? 

(t  i....u...i'...-i.,y{?..i!..i...i...i.,??!ti  i  1 1    ii»^,J.iJl,.l.,.i.,.i.,'W.rrT! 

FIG.  56. 

engraved  millimetre  scale,  on  which  an  index  of 
mercury  (5)  shows  any  change  in  the  volume  of  the 
enclosed  column  of  air  (aft).  Before  closing  the 
end  of  the  tube  (a),  the  tube  should  be  thoroughly 
cleaned  and  dried. 


120 


EXPANSION  OF  GASES. 


[Exp.  26. 


To  test  the  calibre  of  the  tube,  we  first  weigh  it 
when  empty;  then  we  pour  in  some  pure  mercury 
(see  ^  13)  to  a  depth,  let  us  say,  of  5  cm.,  working 
it  well  into  the  bottom  of  the  tube  by  means  of  a  fine 
steel  wire.  The  depth  of  the  mercury  is  then  found 
as  accurately  as  possible  by  the  millimetre  scale,  and 
the  tube  is  re-weighed.  Then  more  mercury  is 
added,  a  little  at  a  time.  After  each  addition,  the 
depth  is  recorded,  and  the  corresponding  weight  is 
found.  This  process  is  continued  until  the  tube  is 
nearly  filled  with  mercury,  when  the  calibration 
is  complete. 

Subtracting  from  each  weighing  that  of  the  empty 
tube,  we  find  the  amount  of  mercury  contained  at 
each  step  in  the  process.  Multiplying  each  weight 
of  mercury  in  grams  by  the  space  in  cu.  cm.  occupied 
by  each  gram  (0.0738  at  20°)  we  have  the  capacity 
of  the  tube  corresponding  to  the  different  depths  ob- 
served. The  results  are  to  be  entered  on  co-ordinate 
paper  in  the  usual  method 
(§  59).  Thus  in  Fig.  57 
the  crosses  represent  volumes 
from  *1  to  '7  cu.  cm.  corre- 
sponding to  depths  from  0  to 
50  cm.  The  curve  enables 
us  to  find  the  volume  of  air 
enclosed  by  the  index  of  mer- 
cury (6,  Fig.  56)  at  any  point  of  the  tube.  It  is  easy 
to  show  by  geometry  that  unless  the  crosses  all  lie  in 
the  same  straight  line,  the  tube  cannot  be  of  uniform 
calibre. 


•10  -20  -30  -»0  -50  -60  -70 


FIG.  57. 


T  72.]  THE   AIR  THERMOMETER.  121 

^|  72.  Precautions  in  the  Use  of  an  Air  Thermometer. — 
To  obtain  accurate  results  with  an  air  thermometer, 
it  is  necessary  that  the  tube  should  be  perfectly 
clean  ;  for  any  foreign  matter  may  interfere  with  the 
free  motion  of  the  mercury  index.  If  in  the  process 
of  calibration  the  tube  has  become  coated  with  the 
impurities  which  mercury  sometimes  contains,  it 
should  be  scoured  with  a  small  wad  of  cotton  on  the 
end  of  a  fine  steel  wire.  Moisture  in  the  tube  must 
be  avoided  with  the  utmost  care,  on  account  of  the 
vapor  which  it  generates  when  heated  ;  and  in  case 
the  slightest  trace  of  condensation  appears,  the 
tube  must  be  heated,  and  dried  by  a  current  of  air 
conducted  through  a  still  finer  tube  to  the  very 


FIG.  58. 

bottom  of  the  thermometer.  The  tube  must  be  large 
enough  to  allow  a  free  motion  to  the  mercury  index, 
but  not  so  large  that  bubbles  of  air  may  force  their 
way  through  the  mercury. 

The  mercury  used  should  be  of  the  purest,  —  at 
least  twice  distilled,  and  perfectly  clean  and  dry.  It 
may  be  introduced  into  the  tube  by  means  of  a  medi- 
cine dropper  drawn  out  in  a  flame  so  as  to  have  a 
long  fine  point  (Fig.  58).  By  piercing  the  mercury, 
as  in  Fig.  59,  and  inclining  the  tube,  the  position  of 
the  globule  may  be  varied  at  pleasure.  It  will  be 
found  convenient  to  place  the  index  so  that  the  lower 
end  may  point  to  a  number  on  the  millimetre  scale 


122  EXPANSION  OF  GASES.  [Exp.  26. 

corresponding  to  the  "absolute  temperature"  (§  76). 
Thus  if  the  temperature  of  the  room  is  20°,  the  lower 
end  may  be  placed  at  a  distance  of  273  +  20,  or 
293  mm.  from  the  bottom  of  the  tube.  "  Absolute 
temperatures "  are  indicated  approximately  *  by  an 
air  thermometer  thus  constructed ;  but  as  the  ther- 
mometer is  affected  by  barometric  changes  as  well  as 
by  changes  in  temperature,  the  indications  should 
always  be  corrected  by  the  method  explained  in  the 
next  section. 

To  eliminate  the  effect  of  the  weight  of  the  index, 
the  experiment  should  be  arranged  so  that  the  air 
thermometer  may  be  observed  always  in  the  same 
position.  It  is  necessary,  also,  that  the  whole  col- 


FIG.  59. 

umn  of  air,  as  far  as  the  index,  should  be  heated  or 
cooled  to  the  temperature  which  is  to  be  measured. 
The  index  must  therefore  be  partly  covered  in  many 
observations  by  the  heating  or  cooling  apparatus,  so 
that  an  observation  of  the  upper  or  outer  end  will 
alone  be  possible.  In  such  cases  the  length  of  the 
index  must  be  allowed  for,  as  what  we  wish  to  find  is 
the  space  occupied,  not  by  the  air  and  the  mercury 
together,  but  by  the  air  alone.  The  length  of  the 
index  must  be  found  by  a  separate  observation  in 
each  case,  as  it  is  not  necessarily  the  same  in  different 
parts  of  the  tube. 

1  Within  a  few  degrees.  The  air  thermometer  here  described  is 
affected  to  the  extent  of  about  4°  for  a  rise  or  fall  of  1  cm.  in  the 
barometer. 


1  73.]  THE  AIR  THERMOMETER.  123 

^|  73.  Determination  of  Temperature  with  an  Air 
Thermometer.  —  The  reading  (r)  of  an  air  thermome- 
ter is  observed,  let  us  say,  in  a  horizontal  position, 
and  compared  with  that  of  a  mercurial  thermometer 
beside  it.  The  air  thermometer  is  then  surrounded 
in  a  horizontal  trough  by  melting  snow  or  ice,  and  the 
reading  (r)  of  the  lower  end  of  the  index  either  di- 
rectly or  indirectly  determined  (see  IF  72).  Then  it 
is  surrounded  by  steam,  in  an  apparatus  similar  to 
that  shown  in  Fig.  46,  ^f  57,  and  the  reading  (rx)  is 
again  observed.  The  air  thermometer  is  finally  al- 
lowed time  to  cool  to  the  temperature  of  the  room, 
and  again  compared  with  the  mercurial  thermometer. 
We  will  assume,  in  the  absence  of  any  marked 
change  in  the  barometer  or  in  the  temperature  of 
the  room,  that  the  air  thermometer  returns  to  its 
original  reading,  r  ;  if  it  does  not,  the  experiment 
should  be  repeated. 

Referring  to  the  curve  found  in  the  calibration  of 
the  tube  (Fig  57,  ^f  71),  we  now  find  the  volumes 
v,  v0,  vv  of  the  confined  air  corresponding  respec- 
tively to  the  observed  readings,  r,  r0,  r15  of  the  lower 
end  of  the  index.  The  temperature  (f)  indicated 
by  the  air  thermometer  is  then  calculated  by  the 
formula 


which  is,  however,  strictly  accurate  only  when  the 
barometer  stands  at  76  cm.  (see  ^[  74,  VIII.).  It 
is  interesting  to  compare  the  reading  of  a  mercurial 
thermometer  with  the  true  temperature  as  indicated 


124  EXPANSION  OF  GASES.  [Exp.  26. 

by  an  air  thermometer,  even  if  (as  will  probably  be 
the  case)  the  accuracy  of  the  observations  will  not 
justify  a  correction  of  the  mercurial  thermometer.1 
Instead  of  air,  coal-gas  or  hydrogen  may  be  em- 
ployed in  a  thermometer,  or  in  fact  any  gas  not 
easily  liquefied.  The  results  are  essentially  the  same 
as  with  the  air  thermometer.  At  the  same  time  that 
air  thermometers  have  for  various  reasons  (see  ^[  74) 
been  adopted  as  standards  of  temperature,  it  is  found, 
by  carefully  comparing  them  with  mercurial  ther- 
mometers, that  the  difference  in  their  indications  at 
ordinary  temperatures  is  generally  small  in  compari- 
son with  errors  of  observation.  On  account  of  their 
greater  convenience  and  precision,  mercurial  ther- 
mometers are  therefore  employed  in  most  scientific 
determinations. 

^[  74.  Theory  of  the  Air  Thermometer.  —  The  air 
thermometer  depends  upon  the  Law  of  Charles 
(§  80),  that  the  volume  of  a  gas  under  a  constant 
pressure  is  proportional  to  its  "  absolute  tempera- 
ture "  (§  76)  ;  that  is,  to  its  temperature  when  reck- 
oned from  a  certain  point,  about  273°  centigrade 
below  freezing,  at  which  it  is  supposed  that  all  sub- 
stances would  be  completely  devoid  of  heat.  If 
jP,  T0,  and  2\  represent  respectively  the  absolute  tem- 
perature at  which  the  volumes  v,  w0,  and  vt  were  ob- 
served, we  have,  according  to  the  law  stated  above, 

T,  :  T9  :  :  vt  :  v0  I. 

T:T0::v:v0  II. 

1  To  lend  interest  to  this  experiment,  the  student  may  be  provided 
with  a  very  inaccurate  mercurial  thermometer. 


174.]  THE   AIR   THERMOMETER.  125 

From  I.  and  II.  we  find  by  one  of  the  ordinary  rules 
of  proportion, 

^-T0  =  v,-vQ 
TO  v0 

"-^'- 

Dividing  IV.  by  III.  we  have 

T-T0_v-v0 
2\-ro  -,,-,; 

Now  the  difference  between  the  freezing  and 
boiling  temperatures,  Tt  and  T0,  under  the  normal 
barometric  pressure  (76  cm.)  is  divided  on  the  centi- 
grade scale  into  100  parts,  called  degrees,  or 

Z\  —  T0  =  100°,  VI. 

and  any  ordinary  temperature,  £,  is  measured  by  the 
excess  of  the  corresponding  absolute  temperature 
(T)  above  the  freezing  point  (  jP0)  ;  that  is, 

T—  T0  =  t.'  VII. 

Substituting  the  values  of  T^  —  T0,  and  T—  T0  in  VI. 
and  VII.  for  their  equivalents  in  V.,  and  multiplying 
by  100°,  we  have  (at  76  cm.  pressure), 

f  =  100°-^=^0.  VIII. 


If  the  barometer  does  not  stand  at  76  cm.  we  substi- 
tute for  100°  in  the  equation  the  actual  number  of 
degrees  between  freezing  and  boiling  (see  Table  14). 
The  student  may  test  the  accuracy  of  his  work  by 
calculating  the  "  absolute  zero  "  (2),  in  this  case,  the 
temperature  at  which  the  index  would  reach  the 


126  EXPANSION  OF   GASES.  [Exp.  26. 

bottom  of  the  tube,  provided  that  there  were  no 
change  in  the  rate  at  which  the  air  contracts.  Sub- 
stituting in  equation  VIII.  v  =  0,  we  have  at  76  cm. 
pressure, 

z  =  —100°^—,  IX. 

VI—VQ 

in  which  the  factor  100°  should  strictly  be  corrected 
as  in  VIII.  for  barometric  pressure.  The  meaning 
of  this  equation  is  particularly  evident  in  a  special 
case.  If,  for  example,  in  a  perfectly  uniform  tube, 
the  index  falls  from  a  reading  of  373  mm.  in  steam 
to  a  reading  of  273  mm.  in  ice,  —  that  is,  100  mm. 
for  100°,  or  1  mm.  per  degree,  —  it  is  clear  that  to 
reach  the  bottom  of  the  tube  it  must  traverse  still 
farther  a  distance  of  273  mm.,  corresponding  to  273° 
of  the  same  length.  The  result  of  this  experiment, 
when  accurately  performed  with  any  of  the  so-called 
"  permanent  gases "  is  invariably  to  indicate  a  tem- 
perature not  far  from  — 273°  C.  for  the  absolute 
zero.  It  is  evident  that,  if  the  volume  of  a  gas 
contracts  by  an  amount  equal  to  one  273d  part  of 
its  volume  at  the  freezing-point  for  every  degree 
which  it  is  cooled,  the  volume  will  be  reduced  to 
nothing  at  the  temperature  of  273°  below  zero ;  and 
conversely,  if  z  is  the  absolute  zero,  that  the  gas 
must  gain  or  lose  one  zth  part  of  its  volume  at  zero 
degrees  when  it  is  heated  or  cooled  1°  centigrade. 
The  coefficient  of  expansion  (e)  (§  83)  is  therefore 
numerically  equal  to  I-t-z;  and  maybe  calculated 
by  the  formula 


100°  X  t>.- 


THE   AIR  THERMOMETER. 


127 


The  coefficient  of  expansion  of  all  permanent  gases 
is  in  the  neighborhood  of  .00367. 


EXPERIMENT  XXVII. 

THE   AIR   THERMOMETER,    II. 

*[[  75.  Construction  of  an  Absolute  Air-Pressure 
Thermometer.  —  A  form  of  air  thermometer  depend- 
_  ent  almost  entirely  upon  pressure  is 

represented  in  Fig.  60.  It  consists 
of  a  U-tube  (aic),  with  a  large  bulb 
(c)  blown  at  the  end  of  the  shorter 
arm,  and  a  somewhat  smaller  bulb 
(a)  at  the  end  of  the  longer  arm. 
The  apparatus  is  sealed  at  the  at- 
mospheric pressure  with  enough  mer- 
cury to  fill  the  smaller  bulb  more 
than  half-full. 

It  is  evident  that  at 
the  absolute  zero  of 
temperature  (see  §75), 
in  the  absence  of  any 
pressure  in  either  bulb, 
the  mercury  must  stand 
at  the  same  level  in  both 
arms  of  the  U.  To  lo- 
cate the  absolute  zero  accordingly, 
mercury  is  poured  back  and  forth 
from  one  bulb  to  the  other  until  no  difference  in  the 
level  is  observed  when  the  thermometer  is  returned 


FIG.  61. 


FIG.  60. 


128  PRESSURE  OF  GASES.  [Exp.  27. 

to  a  vertical  position.  The  zero  of  a  millimetre  scale 
is  now  adjusted  to  this  level  (see  Fig.  61).  By  pour- 
ing mercury  into  the  bulb  a  (Fig.  60),  and  suddenly 
restoring  the  thermometer  to  an  upright  position,  the 
mercury  in  the  tube  will  be  found  to  stand  above 
its  level  in  the  cistern,  owing  to  the  compression  of 
air  in  c  and  its  rarefaction  in  a.  This  process  is  re- 
peated with  more  or  less  mercury  in  a  until  the 
column  reaches  a  point  b  on  the  scale  corresponding 
to  the  absolute  temperature  (see  ^f  72).  The  ther- 
mometer should  now  indicate  any  temperature  cor- 
rectly on  the  absolute  scale,  and  has  the  advantage 
over  that  employed  in  Experiment  26  of  being  un- 
affected by  atmospheric  pressure. 

In  practice,  the  bulb  c  is  made  so  much  larger  than 
the  tube  (6)  that  no  account  need  be  taken  of  the 
variation  of  the  mercury  level  in  c.  The  height  of 
the  mercurial  column  is  measured  accordingly  by  a 
fixed  scale.  The  expansion  of  the  air  in  the  bulb  c 
is  also  disregarded,  together  with  the  compression  of 
the  air  in  a.  All  these  causes  tend  to  diminish  the 
sensitiveness  of  the  thermometer. 

The  air  thermometer  represented  in  Fig.  60  depends 
upon  the  principle  (§  76)  that  the  pressure  of  a  gas 
which  is  prevented  from  expanding  increases  in  pro- 
portion to  the  absolute  temperature.  When  both 
bulbs  (a  and  c)  contain  gas,  the  pressure  in  each 
increases,  and  hence  also  the  difference  in  pressure 
between  them  increases  with  the  absolute  temper- 
ature. It  follows  that  the  height  of  the  mercurial 
column  which  can  be  maintained  by  the  difference 


IT  76.] 


THE   AIR  THERMOMETER. 


129 


of  pressure  in  question  itself  varies  as  the  absolute 
temperature. 

^[  76.  Determination  of  Temperature  by  the  Pressure 
of  Confined  Air.1 — A  tube  (<?,  Fig.  62),  already  em- 
ployed in  ^[  71,  is  to  be  connected  with  a  mercury 
manometer  (a6)  constructed  as  follows :  two  bottles, 
a  and  5,  are  each  provided  with  two  siphons  passing 
through  an  air-tight  stopper,  one  to  the  top,  the 
other  to  the  bottom  of  the  bottle.  The  long  siphons 


and  a  thick-sided  rubber  tube  connecting  them  are 
filled  with  mercury,  and  enough  more  is  added  to 
fill  both  bottles  half- full.  The  mercury  stands  natur- 
ally at  the  same  level  in  the  two  bottles;  and  without 
disturbing  this  level,  the  tube  c  is  connected  to  the 
short  siphon  of  one  of  the  bottles,  5,  by  a  thick 

1  An  experiment  illustrating  the  increase  of  pressure  produced 
by  temperature  will  be  found  in  Exercise  25  of  the  "Elementary 
Physical  Experiments,"  published  by  Harvard  University. 


130  PRESSURE   OF  GASES.  [Exp.  27. 

rubber  tube,  and  the  reading  of  the  index  determined. 
All  the  joints  must  be  carefully  wound  with  string 
to  prevent  leakage. 

The  tube  c  is  now  surrounded  with  melting  ice, 
which  may  be  contained  in  a  horizontal  trough  (see 
^[  57),  leaving  only  the  outer  end  of  the  mercury 
index  uncovered.  The  position  of  the  index  is  then 
accurately  observed.  A  reading  of  the  barometer  is 
made.  The  tube  (c)  is  next  surrounded  with  steam, 
in  a  steam  jacket  (Fig.  46,  ^[  57).  The  air  within  c 
is  prevented  from  expanding  by  raising  the  bottle,  a, 
on  an  adjustable  platform  to  a  certain  height  above  b 
(see  Fig.  62).  The  height  of  b  is  to  be  adjusted  so 
that  the  mercury  index  in  the  tube  c  may  stand  at 
exactly  the  same  point  as  before.  The  vertical  dis- 
tance between  the  mercury  levels  in  a  and  b  is  then 
measured  with  a  metre  rod.  The  tube  c  is  now 
cooled  by  filling  the  jacket  with  water,  the  tempera- 
ture of  which  is  to  be  found  approximately  by  a 
mercurial  thermometer.  The  height  of  the  bottle,  a, 
is  again  adjusted  so  that  the  index  may  return  to  its 
original  position  ;  and  the  difference  between  the  two 
mercury  levels  is  measured  as  before. 

Let  k0  be  the  height  of  the  barometer,  h}  the  height 
of  mercury  required  to  prevent  the  air  from  expand- 
ing when  heated  to  100°  (nearly),  and  h  the  height 
required  to  confine  it  at  the  (true)  temperature,  t ; 
if  we  call  the  pressures  of  the  air  »„,  vn  and  v  at  the 
absolute  temperatures  Tn,  T^  and  T,  respectively  ;  then 
by  definition  (§  74)  we  have,  as  in  ^f  74,  I.  and  II., 

T,  :  T0::  v,  :  v    and   T:  T  :  :  v  :  v; 


H76.1  THE   AIR  THERMOMETER.  131 

from  which  we  may  find,  as  before,  the  temperature, 
t  (^[  74,  VIII.),  the  absolute  zero,  z  (^  74,  IX.), 
and  a  coefficient,  e  (^f  74,  X.),  which  determines  in 
this  case  the  proportion  in  which  the  pressure  of  con- 
fined air  increases  when  heated  1°  centigrade.  Sub- 
stituting the  values  of  v0,  vv  and  t>,  we  find 

t  =  !QQ°~    z  =  ~  100°  t°     e—    h* 

hi  *i  100°  h0 

It  is  believed  that  in  the  case  of  a  perfect  gas  the 
coefficient  which  determines  the  increase  of  pressure 
per  degree  should  be  the  same  as  the  coefficient  of 
expansion  (Experiment  26).  In  practice,  differences 
are  observed  even  with  the  most  permanent  gases  ; 
but  these  differences  are  small  in  comparison  with  the 
errors  of  observation  which  the  student  is  likely  to 
make. 

It  is  interesting  to  compare  the  temperature,  £,  in- 
dicated by  an  air-pressure  thermometer  with  that 
indicated  by  a  mercurial  thermometer,  and  to  test 
the  accuracy  of  the  work  by  calculating  the  tempera- 
ture (z),  at  which  air  would  be  wholly  devoid  of 
pressure,  as  well  as  the  coefficient  e,  relating  to 
change  of  pressure.  If  the  results  agree  with  the 
values  given  in  ^[  74,  within  one  or  two  per  cent,  the 
student  will  be  justified  in  applying  a  correction  to 
the  mercurial  thermometer. 


132  PRESSURE  OF  VAPORS.  (Exp.  28. 

EXPERIMENT    XXVIII. 

PRESSURE   OP   VAPORS,   I. 

^f  77.  Application  of  the  Law  of  Boyle  and  Mariotte 
in  the  Air  Manometer.  —  One  of  the  most  important 
applications  of  the  Law  of  Boyle  and  Mariotte 
(§  79)  is  in  the  construction  of  a  pressure-gauge,  or 
manometer.  A  simple  form  is  represented  in  Fig.  62. 
It  consists  of  a  U-tube,  closed  at  one  end 

\\    (l        an(*  ^e(*  w*tn  mercury  UP  to  a  certain 

i JJ    y        level,  corresponding  to  No.  1  on  the  gauge. 

The  open  end  of  the  U-tube  is  connected 

*  fej        with  the  interior  of  a  vessel,  the  pressure 

IG'  '  in  which  is  to  be  determined.  If  the  mer- 
cury stands  as  before  at  No.  1,  we  know  that  the 
vessel  must  be  at  the  ordinary  atmospheric  pressure. 
If,  however,  the  air  in  the  closed  arm  is  compressed 
to  half  its  original  volume,  we  know  that  the  pressure 
must  amount  to  2  atmospheres  ;  if  the  air  is  reduced 
to  one-third  its  original  volume,  the  pressure  is  3 
atmospheres,  etc.  If,  on  the  other  hand,  the  air 
expands,  the  pressure  must  be  less  than  1  atmo- 
sphere. The  pressure  in  atmospheres  may  therefore 
be  indicated  directly  on  a  scale  properly  spaced. 
No.  2  is,  for  instance,  half-way  between  the  closed 
end  of  the  tube  and  No.  1 ;  No.  3  is  one-third  way  ; 
No.  4  one-quarter  way,  etc.  Such  a  gauge  is  useful 
in  experiments  where  it  is  necessary  to  know  roughly 
the  pressure  in  a  closed  vessel,  as,  for  instance,  a 


IF  78.]  THE  AIR  MANOMETER.  133 

steam  boiler.  When  accuracy  is  desired,  it  is  neces- 
sary to  increase  the  length  of  the  tube,  to  calibrate  it 
(see  ^[  71),  and  to  allow  for  the  hydrostatic  pressure 
of  the  liquid  in  the  bend. 

The  tube  already  calibrated  (^[  71),  for  the  purpose 
of  measuring  the  expansion  of  air,  may  serve  as  a 
manometer.  The  manometer  may  be  surrounded  (if 
necessary)  with  water,  to  prevent  the  temperature 
from  varying  perceptibly  in  the  course  of  the  ex- 
periment. 

^[78.  Testing  an  Air  Manometer.  —  The  tube  (<?) 
is  to  be  connected,'  as  in  ^[  76,  with  the  bottle  b 
(Fig.  62),  and  the  reading  of  the  index  determined. 

When  the  bottle  a  is  raised,  by  means  of  an  adjust- 
able platform,  above  the  bottle  6,  the  air  in  5,  and 
hence  that  in  c  will  be  subjected  to  a  pressure 
which  can  be  determined  by  measuring  the  dis- 
tance between  the  two  mercury  levels  in  a  and  in  b 
by  means  of  a  vertical  metre  rod  (see  Fig  62).  The 
reading  of  the  manometer  c  is  again  determined. 
The  bottle  b  is  now  raised  above  a,  so  that  the  air 
in  b  and  hence  also  in  c  will  be  rarefied  by  an  amount 
determined  in  the  same  way  as  before.  To  find  the 
original  pressure  in  c,an  observation  of  the  barometer 
is  made  (^  13). 

Let  h  be  the  height  of  the  barometer,  \  that  of  the 
column  (a6)  producing  compression,  h.2  that  produ- 
cing rarefaction  ;  and  let  the  corresponding  volumes 
of  air  enclosed  by  the  index  in  c  be  respectively 
(see  ^[  71,  Fig.  57)  v,  v^  v2,  at  the  pressures^,  p^  p2; 
then  evidently  p  =  h;  pt  =  A-j-Aa;  p2  =  h  —  A,. 


134  PRESSURE   OF   VAPORS.  [Exp.  28. 

Now,  according  to  the  law  of  Boyle  and  Mariotte 

(§  79), 

vp  =  v1p1  =  v2pi; 

hence  we  should  find 

v  X  h  =  vl  X  (A+&0  =v*X  (h  —  A2). 
If  these  products  differ  by  an  amount  greater  than 
can  be  attributed  to  errors  of  observation,  the  deter- 
minations upon  which  they  depend  should  be  repeated 
before  making  use  of  the  manometer.1 

IF  79.  Determination  of  the  Pressure  of  a  Vapor  by 
an  Air  Manometer.  —  The  air  manometer  which  has 
just  been  tested,  is  first  read  at  the  atmospheric 
pressure,  then  connected  with  a  thick  rubber  tube  to 


FIG.  64. 

a  stout  tube  of  glass,  closed  at  one  end,  and  contain- 
ing ether,  already  boiling  (Fig.  64).  The  boiling 
may  be  effected  with  safety  2  by  hot  water,  between 
50°  and  60°.  The  manometer  should  be  horizontal, 
but  raised  somewhat,  so  that  the  ether  condensing  in 
the  rubber  tube  may  run  back  into  the  boiler.  As 
soon  as  the  ebullition  is  checked  by  the  pressure  of 
the  vapor  generated,  an  observation  of  the  manome- 
ter is  made  ;  and  at  the  same  time,  as  nearly  as  pos- 

1  In  testing  an  air  manometer  from  |  to  2  atmospheres,  the  errors 
due  to  departure  from  the  Law  of  Boyle  and  Mariotte  will  not  amount 
to  one  fourth  of  one  per  cent. 

3  On  account  of  the  danger  of  fire,  all  flame  should  be  removed 
from  the  immediate  neighborhood. 


f  80.] 


LAW  OF  BOYLE   AND   MARIOTTE. 


135 


sible,  the  temperature  of  the  water  is  accurately  re- 
corded. When  the  water  has  cooled  5°,  10°,  etc., 
new  observations  of  the  manometer  are  made.  If  the 
ether  ceases  to  boil,  the  rubber  tube  should  be  cooled, 
or  air  let  out  of  it.  It  is  well  to  put  fresh  ether  in  the 
boiler  from  time  to  time.  The  results  are  accurate 
only  so  long  as  boiling  continues. 

The  pressure,  plt  corresponding  to  any  reading  of 
the  manometer  at  which  the  volume,  vlt  of  air  is  en- 
closed, may  be  calculated  from  the  volume,  v,  at  the 
atmospheric  pressure,  p,  by  the  formula  expressing 
the  Law  of  Boyle  and  Mari- 
otte  (§79), 

_  v 

**      .9* 

The  results  are  to  be  plotted 
on  co-ordinate  paper,  as  ex- 
plained in  §  59,  and  a  curve  FIG  6g 
drawn,   as  in  Fig.  65,   to  il- 
lustrate the  pressure  of  the  vapor  at  various  tem- 
peratures. 


EXPERIMENT  XXIX. 

PRESSURE   OF    VAPORS,    II. 

*[[  80.  Dalton's  Law.  —  We  have  seen  in  the  last 
Experiment  that  the  vapor  of  a  liquid  may  exert  a 
pressure  either  greater  or  less  than  that  of  the  at 
mosphere,  according  to  the  temperature  at  which  the 
liquid  is  maintained.  The  pressure  of  a  volatile 
liquid  is  measurable  even  at  the  ordinary  temperature 


136  PRESSURE  OF  VAPORS.  [Exp.  29. 

of  the  room.  To  prove  this,  one  has  only  to  inject 
a  few  drops  of  ether  with  a  medicine-dropper,  properly 
bent  (see  Fig.  66),  into  the  tube  of  a  barometer  con- 
structed as  in  1[  13.  The  ether  will  form 
bubbles  of  vapor  even  before  it  rises  to  the 
top  of  the  mercurial  column  ;  and  the  pres- 
sure of  this  vapor  will  cause  the  barometer  to 
fall  some  thirty  or  forty  centimetres.  By 
measuring  the  fall  thus  produced,  the  pres- 
sure of  the  vapor  of  various  liquids  at 
different  temperatures  may  be  determined. 

Another  way  to  illustrate  the  pressure  ex- 
erted by  the  vapor  of  a  liquid  is  to  pour  a 
little  of  the  liquid  into  a  flask,  so  that  it  may 
evaporate  into  the  air  which  the  flask  con- 
tains. If  the  flask  is  corked  tightly  as  soon 
as  the  liquid  is  poured  in,  a  considerable  pressure 
may  be  generated.  In  fact,  explosions  sometimes 
occur  from  this  cause.  To  measure  the  pressure,  a 
tube  may  be  passed  through  the  cork  into  some  mer- 
cury in  the  bottom  of  the  flask  (see  Fig.  67),  and  the 
liquid  should  be  injected  by  means  of  a 
medicine-dropper  passing  through  the  cork 
beside  this  tube,  so  as  to  avoid  losing  the 
pressure  generated  by  evaporation  before 
the  cork  can  be  put  into  its  place. 

It  has  been  found  by  experiment  that 
the  quantity  of  liquid  which  evaporates  in 
a  flask  already  containing  air,  and  the 
pressure  which  it  generates,  are  exactly  the  FlG  67 
same  as  in  a  space  from  which  the  air  has 
been  completely  exhausted.  This  discovery  (known 


T81.]  DALTON'S  LAW.  137 

as  Dalton's  Law)  is  believed  to  show  that  the  mole- 
cules of  a  gas  occupy  very  little  space  in  comparison 
with  the  space  between  them,  into  which  a  liquid 
may  evaporate.  In  any  case,  the  height  to  which 
the  mercury  column  is  raised  in  Fig.  67  is  the  same 
as  its  depression  in  Fig.  66,  other  things  being  equal. 
We  shall  make  use  of  this  fact  to  determine  roughly 
the  pressure  of  a  vapor  at  various  temperatures. 

We  have  seen  that  when  a  liquid  evaporates  into 
a  confined  space  filled  with  air,  the  pressure  of  the 
air  is  increased.  It  is  evident  that  in  an  open  flask 
the  air  must  expand  until  the  combined  pressure  of 
the  air  and  the  vapor  inside  becomes  equal  to  the 
atmospheric  pressure  outside.  If  therefore  we  know 
the  pressure  of  the  air  within  the  flask,  and  that  of 
the  air  outside  of  it,  the  difference  must  be  equal 
to  the  pressure  of  the  vapor  in  question.  To  find  the 
pressure  of  the  air  within  the  flask,  it  is  necessary  first 
to  absorb  or  to  condense  the  vapor  which  it  contains. 

^[  81.  Determination  of  the  Pressure  of  a  Vapor  in  the 
Presence  of  Air.  —  To  find  the  pressure  of  aqueous 
vapor  in  an  open  flask,  a  small  quantity  of  water  is 
heated  in  it  by  submerging  the  flask  up  to  the  neck 
in  a  jar  of  hot  water.  The  temperature  of  the  water 
within  the  flask  is  now  determined  by  means  of  a 
thermometer,  and  a  rubber  cork  is  tightly  inserted. 
When  the  flask  has  become  sufficiently  cool  it  is 
weighed,  then  inverted,  opened  under  ice-water, 
corked,  dried,  and  reweighed  with  the  water  which 
enters  it.  Finally,  it  is  filled  with  water  and  weighed 
again.  A  reading  of  the  barometer  is  made. 


138  PRESSURE   OF  VAPORS.  [Exp.  29. 

Let  wv  w.»  and  wz  be  the  first,  second,  and  third 
weights  in  grams,  t  the  temperature,  and  h  the  ba- 
rometric pressure  in  cm.  within  the  flask  ;  then  the 
capacity  (<?)  of  the  flask  in  cu.  cm.  for  air  or  vapor  is 

c  =  u'3  —  tv±  nearly  ; 

and  since  the  volume  of  air  at  0°  is  nearly  w2  —  w2 
cu.  cm.,  its  volume  (v)  at  t°  is  (see  §  80) 
_  (wt  —  to,)  X  273  +  t 
273 

The  pressure  of  this  air  at  t°  is  v  -r-  c  atmospheres 
(§  79),  or  hv  -r-ccm.  Hence  the  pressure  (/>)  of  the 
vapor1  must  be 


^  82.  Evaporation  and  Boiling.  —  The  student  will 
notice  the  regular  increase  of  the  quantity  of  aqueous 
vapor  in  the  air  as  the  temperature  is  increased,  until 
finally,  as  the  water  approaches  its  boiling-point, 
scarcely  any  air  remains  in  the  flask.  It  is  interest- 
ing to  push  the  experiment  still  further,  and  to  expel 
all  the  air  by  actually  boiling  the  water.  Boiling  may 
be  distinguished  from  evaporation  by  the  presence 
of  bubbles  of  pure  steam.  Unlike  the  bubbles  of 
air  set  free  from  the  water  by  the  application  of  heat, 
the  bubbles  of  steam  may  at  first  completely  condense 
with  a  crackling  sound  before  reaching  the  surface  of 
the  liquid.  When,  however,  the  whole  liquid  is 
raised  to  the  boiling-point,  the  bubbles  expand  as 
they  escape  from  the  liquid,  and  if  the  supply  of  heat 

1  We  neglect  in  this  formula  the  pressure  4.6mm.  of  aqueous 
vapor  at  0°. 


IT  82.]  EVAPORATION  AND  BOILING.  139 

is  sufficient,  furnish  a  steady  current  of  steam  which 
issues  from  the  neck  of  the  flask.     The  stopper  is  in- 
serted before  boiling  has  ceased,  but,  to  avoid  explo- 
sion, not  until  the  source  of  heat  has  been  removed. 
When  the  vapor  is  condensed  by  pouring  cold  water 
on  the  bottom  of  the  flask  (Fig.  68),  ebul- 
lition will  take  place  even  after  the  water 
within  the  flask  is  no  longer  warm  to  the 
touch.    If  the  experiment  has  been  suc- 
cessful, a  peculiar  metallic  sound  will  be 
heard  on  shaking  the  water  in  the  flask. 
This  sound  is  called  the  water-hammer, 
and  indicates  an  almost  total  absence  of 
air.     If  the  flask  is  opened  under  water, 
it  should  be  completely  filled.     If  opened  in  air,  the 
space  not  already  occupied  by  water  will  be  filled 
with  air.     The  student  may  be  interested  to  make 
a    rough   determination    of  atmospheric   density  by 
weighing  the  flask  before  and  after  the  admission  of 
air  (see  ^[  44).     The  capacity  of  the  flask  for  air  is 
found   from   the   quantity  of  water  which  must  be 
added   to  that  already  present  in  order  to   fill  the 
flask  (see  ^[  45).     The  principal  objection  to  a  deter- 
mination of  density  by  this  method  lies  in  the  fact 
that  an  unknown  quantity  of  aqueous  vapor  may  be 
taken  up  by  the  air  which  enters  the  flask.     Its  ad- 
vantage consists  in  the  nearly  perfect  vacuum  which 
is  produced  by  the  condensation  of  aqueous  vapor. 
For  further  illustrations  of  evaporation  and  boiling, 
see  Exercise  22  of  the  "  Elementary  Physical  Experi- 
ments," published  by  Harvard  University. 


140  BOILING  AND  MELTING  POINTS.          [Exp.  30. 

EXPERIMENT  XXX. 

BOILING   AND   MELTING   POINTS. 

^[  83.  Determination  of  Boiling  and  Melting  Points.  — 
The  heater  already  used  to  determine  the  boiling- 
point  of  water  on  a  mercurial  thermometer  may  also 
be  employed  to  find  the  boiling-points  of  other  liquids. 
The  chief  objection  to  this  apparatus  is  the  change 
of  composition  which  results  from  boiling  away  an 
impure  liquid,  owing  to  the  fact  that  the  more  vola- 
tile ingredients  are  the  first  to  escape. 
It  becomes  necessary  to  condense  the 
vapor  before  it  escapes  from  the  spout, 
and  to  make  the  liquid  thus  formed  re- 
turn to  the  boiler.  There  are,  moreover, 
two  practical  objections  to  the  use  of 
such  an  apparatus,  —  the  difficulty  of 
obtaining  a  sufficient  quantity  of  liquid 
to  fill  the  boiler,  and  the  danger  of  fire. 
These  objections  are  met  by  boiling 
the  liquid  in  a  long  test-tube,  as  in  Fig. 
FIG.  69.  gg^  »pjie  vap0r  condenses  on  the  sides 
but  does  not  escape,  and  the  danger  of  fire  is  avoided 
by  the  use  of  hot  water  instead  of  a  flame  as  a  source 
of  heat. 

Alcohol,  for  instance,  will  boil  freely  if  the  test- 
tube  is  plunged  in  water  at  or  near  the  temperature 
of  100°,  since  the  boiling-point  of  alcohol  is  between 
78°  and  80°.  As  the  water  cools  it  may  be  used 
successively  to  find  the  boiling-points  of  chloroform 


t  84.]  BOILING  AND   MELTING  POINTS.  141 

(58°-61°),  bisulphide  of  carbon  (47°-48°),  and  ether 
(35°-37°).  It  is  well  to  have  the  water  about  20° 
warmer  than  the  boiling-point  of  the  liquid  which  is 
to  be  determined. 

The  same  apparatus,  or  one  with  a  shorter  tube, 
may  be  used  to  determine  melting-points.  A  piece  of  a 
paraffine  candle  rnay  be  melted  in  the  test-tube  by  hot 
water;  then,  as  it  begins  to  harden,  the  temperature  is 
observed.  Again,  by  the  use  of  hot  water,  the  paraf- 
fine is  gradually  heated,  and  the  temperature  noted  at 
which  it  begins  to  melt.  Owing  to  impurity  of  the 
paraffine,  certain  constituents  usually  congeal  more 
easily  than  others.  It  has,  therefore,  no  definite  melt- 
ing point.  A  certain  variety  of  commercial  paraffine 
melts,  for  instance,  between  53°  and  57°.  The  results 
are  to  be  corrected  as  explained  below. 

^[  84.  Precautions  and  Corrections  in  Determining 
Boiling  and  Melting  Points.  —  To  prevent  radiation  to 
or  from  the  bulb  of  the  thermometer,  and  to  avoid  all 
danger  of  spattering  (see  ^[  69,  I.),  a  shield  may  be 
constructed  out  of  thin  sheet  brass,  small  enough  to 
fit  into  the  test-tube.  The  bulb  must  not  dip  into 
the  liquid,  but  must  be  surrounded  with  vapor.  The 
level  of  the  vapor  will  be  distinctly  visible  through 
the  sides  of  the  tube.  It  should  reach  a  point  a  little 
beyond  the  end  of  the  mercurial  column  in  the  stem 
of  the  thermometer,  but  must  in  no  case  reach  the 
open  end  of  the  test-tube.  A  slight  escape  of  the 
vapor,  due  to  evaporation,  cannot  be  avoided  ;  but  a 
continuous  current  must  be  instantly  arrested  by 
removing  the  source  of  heat. 


142  BOILING  AND  MELTING  POINTS.         [Exr.  30. 

In  finding  melting-points,  the  bulb  and  stem  of  the 
thermometer  should  be  surrounded  with  liquid  up  to 
a  point  just  below  the  end  of  the  mercurial  column. 
If  the  stem  be  dipped  any  farther  into  the  liquid,  it 
may  become  impossible  to  read  the  thermometer. 

The  student  is  advised  not  to  attempt  the  deter- 
mination of  boiling-points  above  100°  C.,1  on  account 
of  the  danger  of  accidents.  It  may,  however,  be  in- 
structive to  explain  how  a  temperature  above  100° 
can  be  determined  with  a  thermometer  reading  only 
to  100°.  A  thread  of  mercury  not  over  100°  in  length 
is  first  broken  off  and  stored  in  the  expansion  chamber 
(c,  Fig.  51,  ^[  66).  The  thermometer  is  then  tested 
in  steam  (^[  69,  I.).  Its  reading  will  be  somewhat 
above  0° ;  let  us  say  15°.  Then  all  the  readings  of 
this  thermometer  will  be  about  85°  too  low.  It  is 
possible,  therefore,  to  determine  temperatures  up 
to  185°. 

We  should,  however,  remember  that  a  column 
measuring  85°  at  a  temperature  of  100°  will  measure 
more  or  less  than  that  amount,  according  to  the  tem- 
perature in  question.  Let  the  length  of  the  thread 
of  mercury,  in  degrees,  be  I,  and  let  the  temperature 
at  which  this  thread  is  actually  observed  be  t  (100° 
in  the  instance  above)  ;  then  if  £,  is  the  tempera- 
ture to  be  determined,  the  correction  in  degrees  is 
.00018?  (t—  O-  This  follows  from  the  value  of 
the  coefficient  of  expansion  of  mercury  ;  for  if  a  thread 

1  Chloroform  should  be  substituted  for  turpentine  (which  boils  at 
about  160°)  in  the  second  Experiment  in  Physical  Measurement  in  the 
list  published  by  Harvard  University. 


IT  84.]  BOILING  AND  MELTING   POINTS.  143 

1°  long  when  heated  1°  centigrade  expands  by  the 
amount  0°. 00018,  then  a  thread  1°  long  when  heated 
(t  —  Zj)0  would  expand  I  X  (t—t  J  times  as  much. 

Thus  the  correction  in  determining  the  boiling-point 
of  turpentine  (160°)  with  a  thread  85°  long,  broken 
off  and  measured  at  the  temperature  100°  instead  of 
160°,  would  be  .00018  x  85  X  (160  — 100),  or  a  little 
over  0°.9.  Instead,  therefore,  of  adding  85°  to  the 
reading  of  the  thermometer  (let  us  say  74°)  we  should 
add,  strictly,  85°.9,  —  that  is,  the  actual  length  of  the 
thread  of  mercury  at  the  temperature  observed.  In- 
stances have  already  been  given  (^[  65,  (4))  of  errors 
resulting  from  heating  only  the  bulb  of  a  thermometer 
to  a  given  temperature.  The  corrections  in  such 
cases  are  calculated  by  the  rule  given  above.  That 
is,  we  multiply  the  length  of  the  thread  exposed  to 
the  air  by  the  difference  in  temperature  between  the 
air  and  the  bulb  of  the  thermometer,  to  find  the  cor- 
rection which  should  be  applied. 

In  all  determinations  of  temperature,  the  readings 
of  the  thermometer  are  made  to  tenths  of  a  degree 
(^[  68),  and  corrected  by  the  table  already  calcu- 
lated (T  70).  The  boiling-points  of  all  liquids  are 
affected  more  or  less  by  atmospheric  pressure.  A 
reading  of  the  barometer  should  always  accompany 
such  determinations. 


144 


CALORIMETRY. 


[Exp.  31. 


EXPERIMENT   XXXI. 

METHOD   OF   COOLING. 

^[  85.  Determination  of  Rates  of  Cooling.  —  A  Cal- 
orimeter (Fig.  70)  is  usually  constructed  of  two  (or 
more)  metallic  cups,  one  inside  of  the  other.  A 
vertical  section  of  the  calorimeter  is  shown  in  Fig. 
71,  and  a  horizontal  section  in  Fig.  72.  The  inner 
cup,  generally  made  of  thin  brass,  has  its  outer  sur- 
face brightly  polished  to  lessen  radiation;  and  for 
the  same  reason  the  outer  cup  should  be  polished 
inside.  To  prevent  the  conduction  of  heat  from  one 


FIG.  72. 


FIG.  70. 


FIG.  71. 


cup  to  the  other,  the  cups  are  separated  by  pieces  of 
cork,  which  should  be  sharpened  to  a  point,  and  held 
in  place  by  wires.  A  large  flat  cork  serves  to  cover 
both  cups,  and  thus  in  a  great  measure  to  prevent 
loss  of  heat ;  for  if  the  top  of  the  calorimeter  were 
open,  a  considerable  quantity  of  heat  would  be 
carried  away  by  currents  of  air.  In  some  cases 


Ti  85.]  METHOD   OF   COOLING.  145 

a  small  stopper  is  also  used,  to  close  the  inner  cup 
water-tight. 

We  prefer  for  most  purposes  a  calorimeter  de- 
pending (like  that  shown  above)  upon  air  spaces 
for  its  insulation,  to  one  in  which  these  spaces  are 
filled  with  wool,  or  other  non-conducting  material;1 
for  though  air  transmits  more  heat  than  wool,  it 
absorbs  much  less.  The  heat  absorbed  by  insulating 
materials  is  a  continual  source  of  error  in  calorimetry, 
because  there  is  no  simple  way  of  allowing  for  it.  On 
the  other  hand,  the  heat  transmitted  through  the 
sides  of  a  calorimeter  can,  as  we  shall  see,  be  easily 
determined. 

(1)  The  inner  cup  is  to  be  filled  with  hot  water,  be- 
tween 90°  and  100°,  and  the  temperature  of  the  water 
is  to  be  found  by  a  thermometer  passing  through  a 
hole  in  the  cork  cover  (Fig.  71).  The  stirrer  at- 
tached to  the  stem  of  the  thermometer  is  used  to 
keep  the  water  in  continual  agitation  ;  and  a  stopper 
is  employed  to  prevent  any  of  it  from  being  spilled 
over  the  edges  of  the  cup.  Observations  of  tem- 
perature are  made  at  intervals  of  one  minute,2  and 
should  be  continued  until  the  thermometer  indicates 
30  or  40  degrees.  The  temperature  of  the  room  is 
then  observed  ;  and  the  quantity  ol  water  which  has 

1  When  no  allowance  is  to  be  made  for  loss  of  heat  by  the  calori- 
meter, the  use  of  felt  is  to  be  recommended.     See  Experiment  10 
in  the  Descriptive  List  of  Chemical  Experiments  published  by  Har- 
vard University. 

2  A  clock  especially  constructed  to  strike  a  bell  once  a  minute 
will  be  found  serviceable  in  the  determination  of  rates  of  cooling. 
Simultaneous  observations  of  time  and  temperature  may  thus  be 
made  (see  §  28). 

10 


146 


CALORIMETRY. 


[Exp.  31. 


been  used  is  determined  by  weighing  the  calorimeter 
with  and  without  it. 

(2)  The  experiment  is  now  to  be  repeated  with  a 
much  smaller  quantity  of  water,  just  enough,  let  us 
say,  to  cover  the  bulb  of  the  thermometer  and  the 
stirrer.     The  calorimeter  is  to  be  inclined  in  every 
possible  direction  between  the  observations  of  tem- 
perature, so  as  to  bring  the  hot  water   in   contact 
with  every  part  of  the  inner  cup. 

(3)  The  experiment  is  again  repeated   with  the 
same  quantity  of  water  as  in  (2),  but  without  inclin- 
ing the  calorimeter.     The  stirrer  is  to  be  used  as  in 
(1),  but  simply  to  secure  a  uniform  temperature  in 
the  water. 

(4)  Finally,  the   calorimeter  is  to  be  filled  with 
glycerine  or  turpentine,  warmed  by  hot  water  (see 
Tf  83).     The  depth  of  the  liquid,  and   the  method  of 
agitation  should  be  the  same  as  in   (1).     The  tem- 
peratures and  weights  are  to  be  observed  as  before. 

The  results  of  (1), 

10  20  3om^o  j-p  60  yo  .  (2),  (3),  and  (4)  are 
to  be  reduced  as  will 
be  explained  in  ^[^[ 
86-89. 

Tf  86.  Effect  of  the 
Temperature  and  Ther- 
mal Capacity  of  a  body 
on  its  Rate  of  Cooling. 

-T  1  g.    I  O. 

—  (1)   The  results  of 

^[  85  (1)  are  to  be  represented  by  a  curve  (aJ,  Fig. 
73),  drawn  on  co-ordinate  paper  as   in  §  59.     The 


s 

\ 

\ 

ff 

\ 

^ 

ft 
Xjjjg 

\ 

^ 

"^ 

X 

'•-., 

"-. 

Ter» 

l>er. 

& 

rr  o 

f  W. 

*   vo 

om; 

|3ir: 

1  86  (1).]  NEWTON'S  LAW  OF  COOLING.  147 

scale  at  the  top  of  the  paper  corresponds  to  the  num- 
ber of  minutes  which  have  elapsed  since  the  first 
observation  was  taken;  the  scale  at  the  left  of  the 
paper  represents  the  observed  temperature  of  the 
water  in  degrees.  The  temperature  of  the  room 
(22|°)  is  shown  by  the  dotted  line,  which  the  curve 
(«6)  should  approach  as  a  limit,  —  that  is,  without 
ever  reaching  it. 

It  is  advantageous  for  many  purposes  that  the 
scale  of  degrees  at  the  left  of  the  paper  should  repre- 
sent, not  the  temperatures  actually  observed,  but  the 
differences  between  those  temperatures  and  that  of 
the  room  ; l  since  the  rate  of  cooling  depends  upon 
the  differences  in  question  (see  §  89).  If  this  method 
is  adopted,  the  first  observation  should  be  one  about 
50°  above  the  temperature  of  the  room. 

In  any  case  the  student  should  satisfy  himself  that 
Newton's  Law  of  Cooling  (§  89)  is  approximately 
fulfilled.2  Thus  the  calorimeter  may  cool  (see  «5, 
Fig.  73)  between  the  5th  and  the  10th  minute  from 
75°  to  70°,  that  is,  5°  in  5  minutes ;  while  between 
the  50th  and  the  60th  minute  it  may  cool  only  from 
44°  to  40o,  or  4°  in  10  minutes.  In  the  first  case, 
when  the  average  temperature  (72^°)  is  50°  above 
that  of  the  room  (22£°)  we  have  a  rate  of  cooling 
equal  to  1°  per  minute ;  in  the  second  case,  with  an 
average  temperature  (42°)  nearly  20°  above  that 

1  This  method  of  plotting  the  curves  must  be  adopted   if  the 
temperature  of  the  room  varies  considerably  in  the  course  of  the 
experiment  (IF  85). 

2  Departures  of  20%  have  been  observed  in  a  range  of  60°.     See 
Everett's  Units  and  Physical  Constants,  Art.  143. 


148  CALORIMETRY.  [Exp.  31. 

of  the  room,  the  rate  of  cooling  is  |°  per  minute. 
Obviously, 

50:20    ::    1  :  f . 

In  the  same  way,  with  20  grams  of  water  in  the 
calorimeter,  the  rate  of  cooling  should  be  found  to 
vary  in  proportion  to  the  excess  of  temperature  above 
that  of  the  room.  The  rate  of  cooling  is,  however, 
very  different  in  different  cases,  as  it  depends  upon 
the  quantity  of  water  which  the  calorimeter  contains. 
Let  us  next  consider  the  relation  between  this  quan- 
tity of  water  and  the  rate  of  cooling. 

(2)  The  fundamental  principle  underlying  all  de- 
terminations by  the  method  of  cooling  is  that  the 
number  of  units  of  heat  (§  16)  lost  by  a  calorimeter 
per  unit  of  time  is  proportional  to  the  difference  in 
temperature  between  the  inner  and  outer  cups.  It 
does  not,  therefore,  depend  upon  the  contents  of  the 
calorimeter  except  in  so  far  as  the  nature  or  quantity 
of  these  contents  may  modify  the  temperature  of  the 
inner  cup. 

Let  us  first  suppose  that  in  both  experiments, 
^[  85  (1)  and  (2),  the  water  is  agitated  sufficiently 
to  bring  it  in  contact  with  every  portion  of  the 
inner  cup,  so  that  a  perfectly  uniform  temperature  is 
the  result ;  then  if  the  outer  cup  is  unchanged  in 
temperature  the  flow  of  heat  from  one  cup  to  the 
other  corresponding  to  a  given  reading  of  the  ther- 
mometer must  be  in  both  cases  the  same.  How,  then, 
do  we  account  for  the  marked  differences  which  we 
observe  in  the  rates  of  cooling? 


T  86  (2).]  RATES  OF  COOLING.  149 

The  supply  of  heat  in  a  calorimeter  may  be  com- 
pared to  the  quantity  of  water  in  a  leaky  pail.  Given 
the  rate  of  the  stream  flowing  out  of  the  pail,  the 
time  it  takes  for  the  water-level  to  fall  one  inch  is 
evidently  proportional  to  the  horizontal  section  of  the 
pail.  In  the  same  way,  with  a  given  flow  of  heat 
from  a  calorimeter,  the  time  required  for  the  temper- 
ature to  fall  1°  must  be  proportional  to  what  we  call 
the  thermal  capacity  (§  85)  of  the  calorimeter  and  its 
contents. 

It  is  obvious  from  Figure  73  that  with  80  grams 
of  water  the  cup  must  cool  more  slowly  than  with 
20  grams.  In  the  first  case  it  takes,  for  instance 
(see  aft,  Fig.  73),  60  minutes  to  cool  from  80°  to  40° ; 
if  in  the  second  case  only  20  minutes  are  required  to 
cover  the  same  range  of  temperature,  the  natural 
inference  is  that  the  thermal  capacity  in  the  first 
case  is  to  that  in  the  second  case  as  60  is  to  20,  or 
as  3  is  to  1. 

The  thermal  capacity  in  question  is  in  no  case 
simply  proportional  to  the  quantity  of  water  which 
the  calorimeter  contains ;  for  the  inner  cup,  the  ther- 
mometer, and  the  stirrer  all  possess  a  certain  capacity 
for  heat.  We  may  estimate  this  capacity  roughly 
by  the  method  of  cooling.  Let  us  call  it  c.  Then  in 
the  first  case  the  total  thermal  capacity  is  80  -\-  c » 
and  in  the  second  case  it  is  20  -j-  c;  hence  we  have 

80  -f  c  :  20  -f  c   ::    3:1, 

a  proportion  which  can  be  satisfied  only  if  c  =  10. 
We  infer,  therefore,  that  the  calorimeter,  thermom- 


150  CALORIMETRY.  [Exp.  31. 

eter,  and  stirrer  are  together  equivalent,  in  thermal 
capacity,  to  about  10  grams  of  water. 

We  may  assume  provisionally  that  this  inference 
is  correct ;  but  for  accurate  calculations,  we  prefer  a 
determination  of  thermal  capacity  made  as  will  be 
described  in  Experiment  32. 

^[  87.  Calculations  concerning  Loss  of  Heat  by  Cool- 
ing. —  We  have  found  in  the  last  section  (^[  86,  1), 
that  when  a  certain  calorimeter  contains  80  grams  of 
water  at  an  average  temperature  50°  above  that  of 
the  room,  the  rate  of  cooling  is  1°  per  minute.  We 
have  also  found  (^[  86,  2)  that  the  calorimeter  itself 
is  equivalent  in  thermal  capacity  to  about  10  grams 
of  water ;  hence  the  total  thermal  capacity  is  80  -f- 10, 
or  90  units.  The  heat  lost  under  these  conditions  is 
therefore  90  X  1,  or  90  units  per  minute.  Let  us 
now  suppose  that  the  average  temperature  is  only 
1°  above  that  of  the  room,  instead  of  50°;  then  by 
Newton's  Law  (§  89)  the  rate  of  cooling  will  be  -ffe 
of  1°  per  minute ;  hence  the  loss  of  heat  will  be 
90  X  -gV,  or  1'8  units  per  minute. 

It  follows  from  the  fundamental  principle  of  the 
method  of  cooling  (^]  86,  2)  that  the  loss  of  heat  at 
a  given  temperature  is  the  same,  no  matter  what 
substance  or  substances  the  calorimeter  may  contain, 
provided  that  every  part  of  the  inner  cup  is  brought 
in  contact  with  the  mixture.  The  rate  of  flow  cor- 
responding to  difference  in  temperature  of  one  degree 
between  the  inner  and  outer  cups  is  accordingly  an 
important  factor  in  calculations  (see  ^[  93,  3)  relating 
to  loss  of  heat  by  cooling. 


187.]  RATES  OF   COOLING.  151 

Unless  the  calorimeter  is  filled,  as  in  ^[  85  (1),  or 
its  contents  sufficiently  agitated,  as  in  (2),  the  inner 
cup  will  not  be  uniformly  heated  throughout.  When 
a  glass  vessel  is  used  (as  in  Exp.  38),  only  those  por- 
tions nearest  the  liquid  may  be  perceptibly  warmed 
or  cooled  by  it ;  and  even  with  metallic  vessels,  es- 
pecially when  thin,  differences  of  temperature  can 
frequently  be  recognized  by  the  touch.  The  result 
is  a  considerable  diminution  in  the  rate  of  cooling. 
To  estimate  the  effect  in  question,  we  may  utilize 
the  results  of  ^[  85  (3). 

From  these  results  the  curve  ac  (Fig.  73)  is  to 
be  plotted  in  the  same  manner  as  ab  (If  86,  1).  If 
in  both  curves  (as  in  Fig.  73)  the  first  observation 
utilized  is  about  80°,  we  shall  find  a  point  of  inter- 
section, a,  nearly  opposite  80°  and  0  minutes.  We 
may  notice  that  ab  takes  60  minutes  to  fall  from 
80°  to  40°,  while  with  ac  only  30  minutes  are  re- 
quired ;  hence  the  rate  of  cooling  represented  by  ac 
is  twice  as  great  as  in  the  case  of  a£,  so  that  when 
reduced  to  1°  difference  in  temperature,  it  will  be 
5^  of  1°  per  minute.  Now  let  the  weight  of  water 
be  20  grams  ;  then  since  the  calorimeter  is  equivalent 
to  10  grams,1  we  have  a  total  thermal  capacity  of 
30  units.  The  loss  of  heat  is  therefore,  not  1-8,  as 
before,  but  30  X  5%  or  1-2  units  per  minute. 

These  figures  are  sufficient  to  show  the  importance, 
in  the  method  of  cooling,  of  comparing  two  quantities 

1  We  should  remember,  strictly,  that  if  only  a  portion  of  the 
inner  cup  is  heated,  the  thermal  capacity  will  be  somewhat  less  than 
10  units. 


152 


CALORIMETRY. 


[Exp.  31. 


under  exactly  the  same  conditions.  Let  us  suppose 
that  we  were  to  calculate  the  thermal  capacity  of  the 
calorimeter  from  the  results  of  IF  85  (1)  and  (3),  in 
which  the  conditions  are  not  the  same.  Since  the  rate 
of  cooling  is  twice  as  great  in  (3)  as  in  (1),  we  might 
infer  that  the  thermal  capacity  of  the  calorimeter 
with  80  grams  was  twice  that  with  20  grams.  This 
would  make  the  thermal  capacity  of  the  calorimeter 
alone  40  units  instead  of  10  (see  ^[  86,  2). 

^[  88.   Construction  of  a.  Series  of  Temperature  Curves. 
—  From  an  extended  series  of  results1  it  would  be 
possible  to  construct  a  series   of  curves  similar  to 
those     shown    in     Fig. 
to\nu*es  74.     It  is  not,  however, 

necessary  that  each  of 
these  curves  should  be 
the  result  of  observa- 
tion. From  two  of 
them,  the  rest  may  be 
obtained  with  more  or 
less  accuracy  by  differ- 
Fm  7£  ent  processes  of  inter- 

polation. 

Let  acegi  and  abdfh  be  the  two  curves  already 
obtained  (see  Fig.  74),  corresponding  respectively  to 
80  grams  and  to  20  grams  of  water,  and  let  it  be 
required  to  draw  a  curve  corresponding  to  50  grams 
of  water.  Then  since  50  is  midway  between  80  and 

1  The  teacher  may,  for  the  sake  of  illustration,  have  a  series  of 
curves  constructed  from  the  results  of  a  large  class  of  students  using 
different  quantities  of  water. 


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nV 

T88.J  CURVES  OF   COOLING.  153 

20,  the  curve  in  question  may  be  placed  (roughly) 
midway  between  the  other  two ;  and  in  the  same 
way  other  curves  may  be  drawn  so  as  to  divide  the 
distance  equally  into  still  smaller  parts.  This  method 
of  interpolation  is,  however,  obviously  inaccurate, 
and  especially  so  between  such  wide  limits. 

A  more  accurate  method  depends  upon  the  prin- 
ciple (see  ^[  86,  2)  that  the  time  of  cooling  is  (other 
things  being  equal)  proportional  to  the  thermal  ca- 
pacity of  the  calorimeter  and  its  contents.  Since 
80  grams  require,  for  instance,  10  minutes  to  cool 
from  80°  to  70°,  and  20  grams  take  only  five  min- 
utes (see  Fig.  74),  we  may  infer  that  50  grams 
would  require  7^  minutes;  or  in  other  words,  that 
the  distance  be  would  be  bisected  by  the  50-gram 
curve.  In  the  same  way  the  other  horizontal  dis- 
tances, de,fg,  hi,  etc.,  would  be  bisected.  To  obtain 
the  intermediate  curves,  accordingly,  the  horizontal 
distances,  be,  de,  fg,  etc.,  are  each  to  be  divided  into 
a  given  number  of  equal  parts.  The  curves  may 
then  be  drawn  through  the  points  of  division. 

It  is  easy  to  show  that  this  method  of  interpolation, 
though  more  accurate  than  the  first,  may  still  lead  to 
considerable  errors,  when  we  consider  differences  in 
the  flow  of  heat  from  the  calorimeter.  With  80 
grams  of  water,  1°  above  the  temperature  of  the 
room,  we  have  calculated  that  the  loss  of  heat 
amounts  to  1-8  units  per  minute  (see  ^[  87)  ;  with 
20  grams  we  have  found  similarly  1  -2  units  per  min- 
ute. Let  us  assume  that  with  50  grams  the  loss  is 
midway  between  these  two  numbers,  or  1-5  units 


154  CALORIMETRY.  [Exp.31. 

per  minute.  Then  since  the  total  thermal  capacity 
is  60  units,  the  temperature  must  fall  at  the  rate  of 
1-5  -T-  60  or^g-of  1°  per  minute.  The  time  required 
to  fall  1°  at  this  rate  would  be  40  minutes ;  in  the 
case  of  80  grams  it  would  be  50  minutes  (see  ^[  87)  ; 
in  the  case  of  20  grams  it  would  be  25  minutes. 
The  times  required  for  80,  50,  and  20  grams  to  fall 
through  a  given  range  of  temperature  would  be, 
accordingly,  proportional  to  the  numbers  50,  40,  and 
25,  respectively.  Since  40  is  by  no  means  midway 
between  50  and  25,  the  50-gram  curve  must  be  con- 
sidered as  only  approximately  bisecting  the  horizontal 
distance  between  the  other  two. 

It  is  evident  that  if  the  system  of  curves  shown 
in  Fig.  74  were  to  be  relied  upon  for  exact  calcula- 
tions, it  would  be  necessary  to  confirm  the  position 
of  the  50-gram  curve,  at  least,  by  direct  observations. 
As  a  matter  of  fact  we  shall  refer  to  Fig.  74  only 
for  the  purpose  of  making  small  corrections  for  cool- 
ing ;  so  that  we  may  disregard  any  errors  in  these 
curves  which  are  likely  to  arise  from  an  interpola- 
tion depending  upon  a  division  of  horizontal  distances 
into  equal  parts. 

^|  89.  Calculation  of  Specific  Heat  by  the  Method 
of  Cooling.  —  I.  A  set  of  curves  is  to  be  constructed 
essentially  as  in  ^[  88,  using,  however,  in  connection 
with  the  curve  acegi  (Fig.  74)  representing  the  re- 
sults of  ^[  85  (1),  a  curve  abdfh,  derived  from  the 
results  of  ^[  85  (2),  and  not  (as  in  Fig.  74)  from 
the  results  of  ^[  85  (3).  The  intermediate  curves 
will  then  represent  rates  of  cooling  corresponding  to 


189.]  CALCULATIONS   FROM   COOLING.  155 

different  quantities  of  water  when  brought  in  contact 
with  every  part  of  the  inner  cup.  The  results  of 
^[  85  (4)  are  next  to  be  plotted  on  tracing-paper, 
with  a  horizontal  line  (as  in  Fig.  73  to)  represent  the 
temperature  of  the  room.  This  line  is  then  super- 
posed (by  moving  the  tracing-paper)  over  a  similar 
line  in  the  new  series  of  curves ;  and  at  the  same 
time  the  curve  on  the  paper  is  made  to  pass  through 
the  common  point  of  intersection  of  the  series  in 
question  (see  a,  Fig.  74). 

A  curve  thus  obtained  with,  let  us  say,  75  grams 
of  turpentine,  may  be  made  to  coincide,  not  with  the 
70-gram  curve,  nor  with  the  80-gram  curve  (see  Fig. 
74),  but  with  one  rather  which  would  correspond  to 
30  or  40  grams  of  water.  Under  the  conditions  of 
the  experiment,  the  heat  lost  by  the  calorimeter  must 
be  the  same  whether  it  contain  turpentine  or  water 
(see  ^[  86,  2)  ;  hence  equal  rates  of  cooling  imply 
equal  thermal  capacities  (ibid.}.  Since  the  calori- 
meter has  the  same  total  thermal  capacity  with  the 
turpentine  as  with  the  water,  the  75  grams  of  turpen- 
tine must  be  equivalent  to  30  or  40  grams  of  water ; 
and  1  gram  of  turpentine  must  be  equivalent  to  a 
quantity  of  water  between  f  §  and  $%  of  a  gram  ;  or 
let  us  say  0.4  -f-  grams.  In  other  words,  the  specific' 
heat  (§  16)  of  turpentine  must  be  0.4+-  In  the 
same  way  the  specific  heat  of  any  other  liquid  might 
be  calculated. 

It  is  evident  that  the  curves  of  ^[  88,  if  thus 
treated,  would  not  have  given  an  accurate  result. 
20  grams  of  water  might  be  found,  for  instance, 


156  CALORIMETRY.  [Exp.  31. 

under  the  conditions  of  ^[85  (3),  to  cool  as  slowly 
as  the  75  grams  of  turpentine  in  ^[  85  (4)  ;  but  this 
would  be  due,  not  simply  to  the  fact  that  water  has 
a  greater  thermal  capacity  than  turpentine,  weight 
for  weight,  but  also  to  the  fact  that  a  much  smaller 
amount  of  surface  is  heated  by  the  water.  Obviously 
the  20  grams  of  water  cannot  be  equivalent  in 
thermal  capacity  to  the  75  grams  of  turpentine,  be- 
cause their  rates  of  cooling,  though  equal,  have  been 
compared  under  dissimilar  conditions. 

II.  Another  method  of  calculating  specific  heat 
depends  upon  a  comparison  of  the  rates  of  cooling 
of  two  liquids  when  equal  volumes  are  employed. 
Let  us  suppose  that  the  time  occupied  by  75  grams 
of  turpentine  in  cooling  from  80°  to  60°  in  ^  85  (4) 
is  really  the  same  as  that  of  20  grams  of  water  in 
^[  85  (3),— that  is,  10  minutes  (see  ac,  Fig.  73),—  while 
that  required  in  ^f  85  (1)  for  80  grams  of  water  (see 
a5,  Fig.  73)  is  20  minutes ;  then  since  the  conditions 
are  nearly  the  same  in  (1)  and  (4),  the  total  thermal 
capacities  in  question  must  be  to  each  other  as  10 
is  to  20  (^[  86,  2).  If  the  calorimeter  is  equivalent 
(see  ^[  86,  2)  to  10  grams  of  water,  we  have  with 
80  grams  of  water  a  total  thermal  capacity  of  90 
units;  hence  with  the  turpentine  the  total  thermal 
capacity  must  be  -|$  of  90  units,  or  45  units.  Sub- 
tracting from  the  45  units  the  10  units  due  to  the 
calorimeter,  we  find  a  remainder  of  35  units,  which 
must  be  the  thermal  capacity  of  75  grams  of  turpen- 
tine. Hence  the  specific  heat  of  turpentine  is  35  -f-  75, 
or  0.4 +. 


IT  90(1).]  THERMAL  CAPACITY.  157 

The  method  of  cooling  has  been  applied  to  the 
determination  of  the  specific  heats  of  solids  in  the 
form  of  powder,  as  well  as  to  liquids;  but  it  is  gen- 
erally thought  to  be  less  reliable  than  the  methods  of 
mixture  about  to  be  described  (Exps.  33  and  34). 


EXPERIMENT  XXXII. 

THERMAL   CAPACITY. 

^[  90.  Determination  of  the  Thermal  Capacity  of  a 
Calorimeter.  —  (1)  We  have  already  seen  that  the 
thermal  capacity  of  a  calorimeter  may  be  calculated 
roughly  from  data  obtained  by  the  method  of  cooling 
(see  ^[  86,  2)  ;  but  that  a  very  slight  change  in  the 
conditions  of  the  experiment  may  make  the  result 
worthless.  For  this  reason  the  method  of  cooling 
is  hardly  to  be  counted  as  a  practical  method  for 
finding  the  thermal  capacity  of  a  calorimeter.  The 
experimental  determination  of  thermal  capacity  may 
be  made  by  either  of  the  following  methods :  — 

I.  The  whole  calorimeter  is  to  be  weighed,  includ- 
ing (see  ^[  85,  Fig.  71)  the  inner  and  outer  cups, 
the  cork  supports  and  cover,  and  the  thermometer 
and  stirrer.  The  temperature  of  the  inner  cup  is 
now  found  by  observing  the  thermometer,  after  it 
has  remained  within  this  cup  for  some  time  (see 
Tf  65,  6).  Then  water  at  an  observed  temperature, 
between  30°  and  40°,  is  poured  rapidly  (^[  92,  4) 
into  the  cup  until  it  is  nearly  full  (f  92,  8).  The 


158  CALORIMETRY.  [Exp.  32. 

cork  is  immediately  inserted  (^[  92,  6)  and  the  time 
noted  (Tf  92,  9).  The  water  is  then  stirred  (Fig.  50, 
^[  65)  by  twisting  the  stem  of  the  thermometer,  until 
two  successive  observations  of  the  thermometer  a 
minute  apart  (see  ^[  92,  10)  agree  as  closely  as  in 
^[  85  (1),  at  the  same  temperature  (see  ^|  92,  8). 
The  resulting  temperature  is  then  observed,  and  the 
time  again  noted  (^j"  92,  9).  The  whole  apparatus  is 
then  re-weighed  to  find  how  much  water  is  in  the 
calorimeter  (see  also  ^f  92,  5). 

There  are  two  practical  objections  to  the  method 
just  described :  first,  that  the  change  in  temperature 
of  the  water  is  almost  too  small  to  be  measured  accu- 
rately with  an  ordinary  thermometer  ;  and  second, 
that  the  quantity  of  heat  absorbed  by  the  calorimeter 
may  be  small  in  comparison  with  that  lost  by  cooling 
(^[  93),  which  can  only  be  roughly  allowed  for. 

The  change  of  temperature  of  the  water  may  be 
increased  by  using  a  smaller  quantity  of  it ;  but  this 
is  objectionable,  as  will  be  seen  by  comparing  the 
results  of  ^[  85,  (2)  and  (3),  unless  the  water  can 
be  well  shaken  in  the  calorimeter,  or  unless  the  object 
of  the  experiment  be  a  determination  of  thermal 
capacity  of  the  calorimeter  when  partly  full.  A  ther- 
mometer graduated  to  tenths  of  degrees  will  be  found 
useful  in  this  and  other  experiments  where  it  is  ne- 
cessary to  measure  small  changes  of  temperature. 

II.  Another  method  of  finding  the  thermal  capa- 
city of  a  calorimeter  consists  in  heating  the  inner 
cup  instead  of  the  water.  This  may  be  done  by  fill- 
ing the  cup  with  hot  lead  (or  better,  copper)  shot, 


f90(2)]  THERMAL   CAPACITY.  159 

the  temperature  of  which  is  to  be  determined  by  two 
or  three  observations  of  a  thermometer  at  intervals 
of  a  minute  (see  ^[  92,  10).  The  shot  must  be  well 
shaken  between  these  observations,  to  secure  a  uni- 
formity of  temperature  (see  ^[  92,  8)  ;  it  is  then 
poured  out,  and  immediately  replaced  by  water  at  an 
observed  temperature  near  that  of  the  room.  The 
resulting  temperature  is  then  determined,  and  the 
weight  of  water  used  is  found  as  before. 

The  change  in  temperature  of  the  water  may  be 
made  practically  five  or  ten  times  as  great  in  II.  as 
in  I.,  and  the  correction  for  its  cooling  will  be  com- 
paratively slight.  The  principal  source  of  error  in 
this  experiment  is  the  rapid  cooling  of  the  inner  cup 
while  empty  (see  ^f  92,  4). 

(2)  The  results  of  an  experimental  determination 
of  thermal  capacity  should  in  all  cases  be  confirmed  by 
a  calculation  based  upon  observations  of  the  weights 
and  specific  heats  of  the  substances  employed  in  the 
construction  of  the  calorimeter.  The  inner  cup  is  to 
be  weighed,  also  the  stirrer  (Fig.  50,  ^f  65)  ;  and  the 
amount  of  water  displaced  by  the  thermometer 
is  to  be  found  by  the  aid  of  a  small  measuring- 
glass  (Fig.  75).  The  glass  should  be  filled 
with  water  so  that  the  thermometer  may  be 
immersed  to  the  same  depth  as  when  it  is 
used  to  determine  the  temperature  of  liquids 
in  the  calorimeter.  The  level  of  the  water  is 
then  carefully  observed  with  and  without  the 
thermometer.  It  will  be  assumed  that  the 
thermometer  is  constructed  of  glass  and  mercury; 


160  CALORIMETRY.  [Exp.  32. 

the  calorimeter  and  stirrer  of  brass;  otherwise  the 
materials  in  question  must  be  noted.  From  these 
data  the  thermal  capacity  of  the  calorimeter  may  be 
calculated  (see  ^  91,  III.)- 

IT  91.  Calculation  of  Thermal  Capacity.  —  We  have 
already  considered  a  method  by  which  thermal  ca- 
pacity may  be  roughly  computed  through  a  compar- 
ison of  rates  of  cooling  (If  86,  2).  This  section 
relates  to  the  calculation  of  thermal  capacity  from 
the  observations  made  in  IT  90. 

If,  as  in  the  first  method  (If  90,  I.),  ^  is  the  orig- 
inal temperature  within  the  calorimeter,  w  the  weight 
of  water  used,  t2  its  temperature  just  before  it  is 
poured  into  the  calorimeter,  and  t  the  resulting  tem- 
perature, then,  since  w  grams  of  water  cool  (t.2  —  £) 
degrees  by  coming  in  contact  with  the  calorimeter, 
they  must  give  up  to  it  w  X  (tz  —  £)  gram-degrees, 
or  units  of  heat  (§  16).  This  raises  the  temperature 
of  the  calorimeter  (t  —  Q  degrees;  hence  to  raise  it 
1°  would  require  a  quantity  of  heat,  c,  given  by  the 
formula 

tpxft-Q  j 

t — £t 

This  is,  by  definition  (§  85),  the  thermal  capacity  of 
the  calorimeter.  To  find  the  temperatures  t  and  £2, 
at  the  time  when  the  water  is  introduced  into  the 
calorimeter,  allowances  for  cooling  should  be  made 
(see  1T^). 

The  second  method  (1"  90,  II.)  differs  from  the 
first  in  that  fy  grams  of  water  are  warmed  (t  —  £>)  de- 
grees, and  hence  must  receive  w  X  (t  —  £2)  units  of 


IT  91.]  THERMAL   CAPACITY.  161 

heat  from  the  calorimeter,  the  temperature  of  which 
is  thereby  reduced  (^  —  t)  degrees ;  hence  to  reduce 
it  1°  would  require  a  quantity  of  heat,  c,  given  by 
the  formula 

•  x(t-Q. 

(«,-<) 

This  formula  is  evidently  reducible  to  the  same 
form  as  I. 

In  the  last  method  (^1  90,  2)  if  w1  is  the  weight  of 
the  inner  cup,  wz  that  of  the  stirrer,  and  ws  the  weight 
(or  volume)  of  the  water  displaced  by  the  thermome- 
ter ;  if  furthermore  st  and  s2  are  the  specific  heats, 
respectively,  of  the  materials  of  which  the  inner  cup 
and  the  stirrer  are  made,1  and  sa  the  thermal  capacity 
of  a  quantity  of  mercury  and  glass  equal  in  volume 
to  a  gram  of  water  ; 2  then  the  thermal  capacity  of 
the  inner  cup  is  u\  s{;  that  of  the  stirrer,  w2  s2;  that 
of  the  thermometer,  w8  s3 ;  hence  the  total  thermal 
capacity  of  the  calorimeter  (c)  is  given  by  the 

formula, 

c  =  wl  s1  +  w2  s2  +  ws  s3.  III. 

If,  for  example,  the  inner  cup  contains  100  g.  of 
brass,  of  the  specific  heat  .094,  its  thermal  capacity  is 

1  The  inner  cup  andtstirrer  are  usually  made  of  brass  (an  alloy  of 
copper  and  zinc),  the  specific  heat  of  which  may  be  taken  as  .094. 

2  It  will  be  noted  that  though  the  specific  heat  of  mercury  (.033) 
differs  greatly   from  that  of  glass   (0.19),  the  thermal  capacity  of 
equal  volumes  is  very  nearly  the  same.     Since  1  cu.  cm.  of  mercury 
weighs  13.6  grams,  it  will  require  13.6  X  .033,  or  0.45  units  of  heat,  to 
raise  it  1°.     In  the  same  way,  since  1  cu.  cm.  of  ordinary  glass  weighs 
not  far  from  2.5  grams,  it  would  require  about  2.5  X  0.19,  or  0.47 
units  of  heat  to  raise  it  1°.     In  calculating  the  thermal  capacity  of  a 
thermometer,  there  will  be,  accordingly,  no  appreciable  error  in  assum- 
ing for  s3  a  mean  value,  0.46. 


162  CALORIMETRY.  [Exp.  32. 

100  X  .094,  or  9.4  units ;  if  the  stirrer  is  made  of 
thin  brass  weighing  2  grams,  its  thermal  capacity  is 
similarly  0.2  units  ;  and  if  the  thermometer  displaces 
0.9  grams  of  water,  its  thermal  capacity  is  (see  2d 
footnote,  page  161)  0.9  X  0.46,  or  about  0.4  units. 
The  total  thermal  capacity  of  a  calorimeter  thus  con- 
structed would  be  9.4  +  0.2  +  0.4  =  10.0  units. 

The  first  method  is  apt  to  give  too  high  results, 
since  the  cooling  of  the  water,  due  to  evaporation 
and  other  causes,  is  attributed  to  contact  with  the 
calorimeter. 

The  second  method  usually  gives  too  low  results, 
on  account  of  the  rapidity  with  which  heat  escapes 
from  the  calorimeter  while  empty.  If,  however,  the 
outer  cup  becomes  heated  indirectly  by  the  shot,  a 
portion  of  this  heat  may  be  radiated  back  to  the 
inner  cup  when  filled  with  water.  It  is  possible, 
therefore,  that  the  results  may  be  too  great. 

The  last  method  generally  gives  too  small  a  re- 
sult, because  we  neglect  the  heat  absorbed  by  the 
materials  surrounding  the  inner  cup.  If,  however, 
only  a  portion  of  the  inner  cup  is  to  be  heated,  we 
may  easily  over-estimate  its  thermal  capacity. 

In  the  latter  case,  we  prefer  an  experimental  deter- 
mination of  thermal  capacity  ;  but  when  the  inner 
cup  is  made  of  very  thin  metal  (as  is  desirable  for 
accurate  work),  the  thermal  capacity  may  be  so 
slight  that  it  cannot  be  exactly  determined  by  ex- 
periment. In  such  cases,  we  usually  depend  upon  a 
calculation  based,  as  in  the  last  method,  upon  the 
weights  and  specific  heats  of  the  materials  composing 
the  calorimeter. 


1T  92  (2).]  PRECAUTIONS.  163 

^[  92.  Precautions  Peculiai  to  Calorimetry. —  In  nearly 
all  experiments  in  calorimetry  two  bodies,  of  known 
weights  and  temperatures,  are  brought  together  so 
that  by  the  flow  of  heat  from  one  to  the  other  (see 
Experiments  33  and  34)  or  by  the  action  of  one  on 
the  other  (see  Experiments  35-38)  a  third  tempera- 
ture results.  There  are,  accordingly,  many  precau- 
tions common  to  these  various  experiments. 

(1)  CHEMICAL  ACTION.  —  It  is  evident  that  the 
substances  employed  should  exert  no  chemical  action 
on  the  sides  of  the  calorimeter.    With  strong  acids,  a 
glass  vessel  should  generally  be  employed.     Instead 
of  a  brass  stirrer,  one  of  platinum  may  be  used.     In 
the   case  of  mercury,  iron  will  do  even  better.     A 
coating  of  asphaltum  is  often  sufficient  to  prevent 
metals  from  being  attacked  by  acids. 

When  two  substances  are  placed  together  in  a 
calorimeter,  neither  should  act  chemically  upon  the 
other  unless  the  object  of  the  experiment  be  to  meas- 
ure the  heat  developed  by  the  reaction.  The  chemical 
relations  between  two  substances  thus  employed  must 
frequently  be  investigated  by  a  separate  experiment. 

(2)  COMPARISON  OF  THERMOMETERS.  —  The  gen- 
eral precautions  necessary  to  the  accurate  observation 
of  temperature  have  been  already  considered  (^[  65), 
and  must  be  observed.     In  addition  to  these  precau- 
tions, certain  others  are  required  when  simultaneous 
observations  of  temperature  are  to  be  made.    In  such 
cases  it  may  be  necessary  to  employ  as  many  ther- 
mometers as  there  are  temperatures  to  be  determined  ; 
and  these  thermometers  have  to  be  compared  with 


164  CALORIMETRY.  [Exp.  32. 

one  already  tested  by  a  process  of  calibration  (^[  68). 
To  do  this,  the  several  thermometers  are  to  be  placed 
in  boiling  water,  in  ice-water,  and  in  water  of  at  least 
three  intermediate  temperatures.  A  large  quantity 
of  water  should  be  used  (see  (3)),  and  it  must  be  well 
stirred  in  each  case.  The  indications  of  each  ther- 
mometer are  to  be  read  in  turn ;  then  again  read  in 
the  inverse  order.  There  should  be  regular  intervals 
(let  us  say  30  seconds  each)  between  the  observa- 
tions. The  two  readings  of  each  thermometer  are 
to  be  averaged,  and  the^  averages  compared.  Know- 
ing (from  Experiment  25)  the  corrections  for  one  of 
the  thermometers,  we  may  easily  calculate  the  cor- 
rections for  the  others.  For  example,  if  three  ther- 
mometers, A,  B,  and  (7,  gave  the  following  readings : 

A,  76°.0  ;  B,  75°.7  ;   <7,  75°.l ;   C,  74°.7  ;  B,  74°.5  ; 
A  74°.0; 

the  average  for  A  would  be  75°.0  ;  for  B,  75°.l ;  for 
O,  74°. 9.  These  averages  evidently  correspond  to 
the  same  point  of  time.  We  should  therefore  sub- 
tract 0°.l  from  the  correction  of  A  at  75°  to  find  that 
of  B  ;  and  we  should  add  0°.l  to  find  that  of  C. 

The  object  of  making  observations  in  the  order 
given  above  is  to  eliminate  errors  due  to  cooling. 

(3)  CONSTANT  TEMPERATURE. —  The  difficulty  of 
making  accurate  observations  of  temperature  at  a 
given  point  of  time  increases  with  the  rate  of  cool- 
ing. The  use  of  large  masses  of  water  (see  (2))  is  one 
of  the  most  general  methods  of  avoiding  rapid 
changes  of  temperature.  In  certain  experiments  in 


T  92  (4).]  PRECAUTIONS.  165 

calorimetry,  special  devices  are  frequently  employed. 
When,  for  instance,  one  of  the  temperatures  to  be 
observed  is  in  the  neighborhood  of  100°,  a  steam- 
heater  may  be  employed  (see  Fig.  77,  also  Fig.  79, 
T[  94).  Again,  a  body  may  be  maintained  at  0°  by 
surrounding  it  with  melting  ice ;  or  it  may  be  kept 
indefinitely,  without  special  precautions,  at  the  tem- 
perature of  the  room,  provided  that  the  latter  be 
constant. 

By  the  use  of  devices  for  maintaining  a  con- 
stant temperature,  thermometric  observations  become 
greatly  simplified.  One  or  more  temperatures  may 
be  known  by  definition,  —  as  in  the  case  of  ice,  or 
steam  at  a  certain  pressure  (§  4).  In  the  absence  of 
cooling,  a  series  of  observations  for  each  temperature 
will  not  be  required,  and  the  temperatures  of  several 
bodies  at  a  given  point  of  time  may  be  found  from 
successive  observations  with  the  same  thermometer. 
The  least  constant  temperature  should  be  observed 
nearest  the  time  in  question. 

(4)  EXPOSURE  TO  THE  AIR.  —  When  a  body  is 
transferred  from  a  heater  or  from  a  refrigerator  to 
a  calorimeter,  there  is  always  more  or  less  heat 
gained  or  lost  from  exposure  to  the  air.  The  time 
of  exposure  should  evidently  be  made  as  short  as 
possible.  In  pouring  liquids,  a  glass  funnel  may  be 
employed  ;  but  the  funnel  must  be  warmed  to  the 
same  temperature  as  the  liquid,  otherwise  it  would 
take  from  it  more  heat  than  the  air.  Water  may  be 
guided  conveniently  from  a  beaker  to  a  calorimeter 
by  a  wet  glass  rod,  ale,  bent  as  in  Figure  76.  To 


166  CALORIMETRY.  [Exp.  32. 

prevent  the  water  from  following  the  side  of  the 
beaker,  the  lip  should  be  greased  at  the  point  b. 
The  wet  stem  of  a  thermometer 
may  also  be  used  as  a  conductor, 
and  with  this  advantage,  that,  since 
the  thermal  capacity  is  easily  found 
(IT  90,  2)  the  heat  required  to  raise 
it  to  a  given  temperature  may  be 
76  calculated.  We  may  notice,  how- 

ever, that  if  the  thermometer  is 
immediately  afterward  placed  in  the  calorimeter,  it 
will  give  up  most  if  not  all  of  the  heat  which  it  has 
absorbed,  and  that  the  remainder  may  be  neglected. 
Hot  shot  may  be  poured  directly  from  a  heater  suit- 
ably shaped  (see  Fig.  79,  ^[  94)  into  a  calorimeter ; 
but  it  is  safer  to  use  a  paper  funnel,  to  prevent  the 
possibility  of  losing  a  portion  of  the  shot.  Most  of 
the  shot  should  enter  the  calorimeter  without  touch- 
ing the  funnel ;  and  the  remainder  should  be  in  con- 
tact with  it  only  for  an  instant.  In  this  case  the  heat 
absorbed  by  the  paper  may  be  neglected.  A  hot  body 
may  also  be  suspended  by  a  thread,  and  thus  trans- 
ferred from  one  place  to  another. 

It  is  obvious  that  the  calorimeter  should  be  brought 
as  near  the  heater  or  refrigerator  as  is  possible  with- 
out danger  that  its  temperature  may  be  affected  by 
radiation,  conduction,  or  convection  from  the  heater 
(§  89).  A  common  pine  board  makes  an  excellent 
shield.  In  Regnault's  apparatus l  (Fig.  77)  the 

1  For  a  fuller  description  of  Regnault's  apparatus,  see  Cooke's 
Chemical  Physics,  page  470. 


T92(4).] 


PRECAUTIONS. 


167 


calorimeter  (at  the  left  of  the  figure)  can  be  brought 
directly  under  the  large  steam  heater  (at  the  right 
of  the  figure).  The  steam  heater  rests  upon  a  sup- 
port, serving  to  shield  the  calorimeter  from  radiation. 
The  support  is  made  hollow,  so  that  it  may  be  kept 
cool  by  a  current  of  water.  The  inner  chamber  of 
the  heater  contains  hot  air.  The  temperature  within 
it  is  observed  by  means  of  a  thermometer  passing 
through  a  cork  by  which  the  top  of  the  chamber  is 


FIG.  77. 

closed.  The  bottom  of  the  chamber  is  closed  by  a 
non-conducting  slide.  By  drawing  the  slide  a  body 
suspended  by  a  thread  in  the  hot-air  chamber  may 
be  lowered  directly  into  the  calorimeter.  The  cal- 
orimeter is  then  immediately  removed  to  a  sufficient 
distance  from  the  heater,  so  that  the  resulting  tem- 
perature may  be  accurately  determined. 

By  devices  similar  to  those  alluded  to,  the  gain 


168  CALOKIMETRY.  [Exp.  32. 

or  loss  of  heat  by  exposure  to  the  air  may  be  almost 
indefinitely  reduced,  but  never  completely  avoided. 
The  student  is  advised  riot  to  attempt  any  correction 
for  this  heat;  because  a  greater  error  might  easily 
result  from  applying  such  a  correction  than  from 
neglecting  it  altogether.  At  the  same  time,  it  is 
well  to  estimate  roughly  the  quantity  of  heat  gained 
or  lost,  with  a  view  to  determining  what  figures  of 
the  final  result  are  likely  to  be  affected. 

For  this  purpose  two  experiments  may  be  made. 
In  one,  a  body  is  transferred  in  the  ordinary  manner 
from  the  heater  or  from  the  refrigerator  to  the  calori- 
meter. In  the  second  experiment,  it  is  passed  back 
and  forth  let  us  say  5  times  each  way,  and  finally 
placed  in  the  calorimeter.  The  body  is  thus  to  be 
exposed  to  the  air  in  one  case  about  11  times  as 
long  as  in  the  other  case,  and  under  similar  condi- 
tions ;  so  that  from  the  difference  in  the  results  we 
may  infer  the  effect  of  an  ordinary  exposure  (see 
IT  93,  4). 

(5)  Loss  OF  MATERIAL.  —  In  rapidly  pouring  a 
liquid  into  a  calorimeter,  or  in  rapidly  lowering  a 
hot  solid  into  a  liquid  already  contained  in  a  calori- 
meter, there  is  danger  that  a  portion  of  the  liquid 
or  solid  may  be  lost.  It  is  accordingly  desirable  to 
weigh,  both  before  and  after  each  addition  to  the 
contents  of  the  calorimeter,  not  only  the  calorimeter 
itself,  but  also  the  vessel  in  which  the  substance  in 
question  was  originally  contained.  The  student  will 
do  well  also  to  make  sure  that  the  space  between  the 
inner  and  outer  cups  is  empty,  both  before  and  after 


Tf  92  (7).]  PRECAUTIONS.  169 

the  experiment ;  for  if  any  of  the  substance  finds  its 
way  into  this  space,  its  loss  will  not  be  apparent 
from  the  weighings. 

(6)  EVAPORATION.  —  A  considerable  portion  of 
the  heat  lost  by  a  liquid  when  poured  into  a  calori- 
meter may  be  caused  by  evaporation.     When  once 
the  liquid  has  been  transferred  to  the  calorimeter, 
all  further  loss  of  heat   by  evaporation  should   be 
prevented  by  immediately  corking  the  inner  vessel. 
It  will  be  assumed  that  the  inner  vessel  is  never 
uncorked,  except  when  necessary  for  the  purposes  of 
manipulation.     Of  two  liquids,  the  denser  is  usually 
the  less  volatile,  and  hence  should  be  heated  in  pref- 
erence to  the  other.     For  the  same  reason,  a  solid 
should  be  heated  in  preference  to  a  liquid.     A  com- 
bustible liquid  should,  as  we  have  seen  (Exp.  30), 
never  be  heated  directly  by  a  flame,  but  indirectly 
by  hot  water. 

(7)  TEMPERATURE  OF  THE  ROOM.  —  The  loss  of 
heat  which  takes  place  from  the  gradual  cooling  of  a 
calorimeter  and  its   contents   depends,  as  we  have 
seen  in  Experiment  31,  upon  the  difference  of  tem- 
perature between  the  inner  cup  and  its  surroundings. 
To  diminish  the  loss  of  heat  in  question,  it  has  been 
proposed  that  the  outer  cup  should  be  placed  in  water 
at  the  same  temperature  as  the  inner  cup.     More  ac- 
curate results  might  be  expected  from  calorimetry  if 
some  means  were  perfected  by  which  we  could  ad- 
just the  temperature  of  surroundings  to  the  needs 
of  an  experiment.     In  practice,  however,  the  experi- 
ment must  be  adapted  to  the  temperature  of  the  air  in 


170  CALORIMETRY  [Exp.  32. 

which  it  is  to  be  performed.  When  considerable  time 
is  required  to  obtain  an  equilibrium  of  -temperature 
(see  (8)  ),  it  is  important  that  the  average  temperature 
within  the  calorimeter  should  agree  as  closely  as 
possible  with  that  of  the  room.  The  weights  and 
temperatures  of  the  substances  employed  in  calor- 
imetry,  are,  therefore,  frequently  chosen  so  as  to  give 
a  final  temperature  between  20°  and  25°. 

It  is  much  easier  to  prevent  than  to  allow  for 
losses  of  heat  by  cooling  ;  and  it  may  be  stated  as  a 
general  rule  in  calorimetry  that  we  must  avoid  in  so 
far  as  possible  all  differences  of  temperature  between 
bodies  under  observation  and  the  objects  by  which 
they  are  surrounded. 

(8)  EQUILIBEIUM  OF  TEMPERATURE.  —  It  has  al- 
ready been  pointed  out  that  to  obtain  a  uniform 
temperature  throughout  the  inner  cup  of  a  calori- 
meter, the  cup  should  be  completely  filled.  If  this 
is  not  done,  special  precautions  must  be  taken  to 
bring  its  contents  into  contact  with  every  portion 
of  its  surface  (see  ^[  85,  2).  The  necessity  of  stirring 
these  contents  has  also  been  alluded  to  (^f  65,  5). 
When  a  mixture  (like  lead  shot  and  water)  is  of 
such  a  nature  that  an  ordinary  stirrer  cannot  be 
used,  the  inner  cup  must  be  closed  water-tight,  so 
that  the  contents  may  be  shaken.  The  thermometer 
should  in  this  case  fit  tightly  into  the  stopper  which 
closes  the  inner  cup,  and  should  reach  into  the  body 
of  the  mixture.  Solids,  if  any  be  used,  should  be 
finely  divided,  so  that  there  may  be  no  risk  of  break- 
ing the  thermometer. 


1 92  (9).]  PRECAUTIONS.  171 

We  prefer,  moreover,  finely  divided  solids,  on 
account  of  the  comparative  rapidity  with  which  an 
equilibrium  of  temperature  may  be  reached,  or  a 
process  of  fusion,  solution,  or  chemical  combination 
completed.  When  a  solid  sinks  in  a  fluid  (as  is 
generally  the  case),  it  is  well  if  it  can  be  warmer 
than  the  fluid,  on  account  of  the  manner  in  which 
convection  currents  are  formed  ;  and  for  the  same 
reason  we  prefer  that  the  denser  of  two  liquids  should 
have  the  higher  temperature.  It  is  always  desirable 
that  the  denser  of  two  substances  should  be  poured 
into  the  other,  so  that,  as  it  passes  through,  as  much 
heat  as  possible  may  be  communicated  from  one  to 
the  other.  The  various  processes  in  calorimetry 
should  in  general  be  completed  in  the  shortest  pos- 
sible time,  especially  when  they  cannot  be  conducted 
at  the  temperature  of  the  room,  since  otherwise  large 
losses  of  heat  are  apt  to  occur. 

Throughout  the  processes  in  question,  stirring  must 
be  interrupted  from  time  to  time,  in  order  that  rough 
observations  of  temperature  may  be  made.  When 
two  successive  observations  agree,  or  when  they' 
differ  by  an  amount  which  may  be  attributed  to  the 
regular  cooling  of  the  calorimeter  (see  Exp.  31), 
the  equilibrium  of  temperature  should  be  complete. 
The  student  will  do  well,  however,  to  make  sure  that 
the  temperatures  at  the  top  and  bottom  of  the  calori- 
meter are  the  same,  before  proceeding  to  make  exact 
observations  of  the  thermometer. 

(9)  TIMING  OBSERVATIONS. — When  observations 
of  temperature  are  taken  regularly  at  intervals  of  one 


172  CALORIMETRY.  [Exp.  32. 

or  two  minutes  throughout  an  experiment,  we  may 
infer  the  time  when  a  given  process  begins  and  when 
it  ends ;  but  to  avoid  errors  due  to  the  possible  omis- 
sion of  one  or  more  observations,  it  is  well  to  note 
the  beginning  and  end  of  each  process  in  hours,  min- 
utes, and  seconds.  In  any  case,  the  time  should  be 
thus  noted,  (1st)  when  all  the  bodies  have  been 
transferred  to  the  calorimeter,  and  (2d)  when,  after 
an  equilibrium  of  temperature  has  been  reached,  the 
resulting  temperature  is  first  observed. 

(10)  SERIES  OP  TEMPERATURES.  —  It  is  well  in  all 
cases  to  make  several  observations  of  the  final  tem- 
perature within  a  calorimeter,  in  order  that  the  result 
may  not  depend  upon  one  alone  (see  §  51).  The 
series  should  be  made  at  intervals  of  one  minute, 
so  that,  as  in  ^f  93  (2),  the  rate  of  cooling  may  be 
found  and  allowed  for.  If  the  calorimeter  contains 
water  only,  we  may  utilize  the  temperature  curves 
already  plotted  (see  ^[  93,  1)  ;  or  if  we  have  deter- 
mined, as  in  ^[  87,  the  flow  of  heat  from  the  calori- 
meter, we  may  make  an  allowance  for  the  heat  lost 
as  in  ^[  93  (3).  In  the  absence  of  any  previous 
determination  under  the  same  conditions  as  in  the 
actual  experiment,  a  series  of  observations  of  the 
temperature  of  the  calorimeter  will  be  required. 

In  the  same  way,  if  the  temperature  of  a  body  is 
changing  perceptibly  before  it  is  placed  in  a  calori- 
meter, it  must  be  determined  by  a  series  of  observa- 
tions. The  intervals  in  all  such  series  would  natur- 
ally be  one  minute  each  ;  but  when  the  temperatures 
of  two  or  more  bodies  are  to  be  found,  the  observa- 


193(1).] 


CORRECTIONS. 


173 


tions  must  be  taken  in  turn.  When  special  precau- 
tions concerning  equilibrium  of  temperature  (see  (8)) 
have  to  be  observed,  the  student  is  advised  not  to 
attempt  observations  at  intervals  of  less  than  one 
minute.  The  temperatures  of  the  several  bodies  con- 
cerned are  to  be  reduced  in  all  cases,  as  in  ^[  93  (1), 
to  the  time  when  they  are  first  enclosed  in  the  calori- 
meter. After  this  time,  losses  of  heat  are  to  be  cal- 
culated as  above,  from  the  known  rate  of  cooling  of 
the  calorimeter. 

Tf  93.  Corrections  for  Cooling.  —  (1)  GRAPHICAL 
METHOD.  —  When  a  calorimeter  contains  water  only, 
as  in  the  determination  Mi  miles, 

of  thermal  capacity 
above  (^[  90,  I.)  or  in 
parts  of  various  experi- 
ments which  follow,  the 
temperature  at  one 
point  of  time  may  be 
inferred  from  an  obser- 
vation taken  at  another 
point  of  time  by  using  one  of  the  curves  in  Fig.  74, 
TT  88.  Let  ab  (Fig.  78)  be  the  curve  corresponding 
to  the  quantity  of  water  which  the  calorimeter  con- 
tains, and  let  c  be  the  observed  temperature.  We 
first  find  a  point  d  on  the  curve  at  the  right  of  c, 
then  a  point  e  above  d.  Then  we  measure  off  a  dis- 
tance efon  the  scale  of  minutes  corresponding  to  the 
length  of  time  during  which  the  calorimeter  has  been 
cooling.  Then  we  find  a  point  g  on  the  curve  below 
/,  and  finally  the  temperature  h,  at  the  left  of  g. 


10     2 

M   soe  40    fo     60    10 

\ 

: 

\ 

s 

; 

\ 

*fl 

N 

< 

\ 

^ 

i 

R» 

£r 

ft& 

r?.« 

f.M 

e  yo 

orn 

ZH* 

FIG.  78. 


174  CALORIMETRY.  [Exp.  32. 

This  temperature  corresponds  in  the  figure  to  a  time 
/  earlier  than  e  ;  but  by  laying  off  the  distance  ef  to 
the  right  of  e,  we  could  find,  if  we  chose,  the  temper- 
ature at  a  later  point  of  time. 

A  more  exact  method  would  be  to  start  with  a 
point  c  (in  Fig.  78),  corresponding  to  a  temperature 
as  far  above  that  of  the  room  (22|°,  Fig.  78)  as  the 
actual  temperature  observed  was  above  the  observed 
temperature  of  the  room.  The  number  of  degrees 
included  between  c  and  6  gives  approximately,  in  any 
case,  the  fall  of  temperature  which  takes  place  in 
an  interval  of  time  corresponding  to  the  number  of 
minutes  between  e  and  /. 

(2)  ANALYTICAL  METHOD.  —  When  several  tem- 
peratures have  been  recorded  at  regular  intervals, 
we  may  infer  the  temperature  at  a  point  of  time 
before  the  beginning  or  after  the  end  of  the  series 
as  follows :  The  observations  are  first  written  down 
in  a  column,  as  in  the  example  below  ;  then  the  tem- 
perature of  the  room  is  subtracted  from  each,  and 
the  results  entered  in  a  second  column  ;  then  a  third 
column  is  formed  from  the  differences  between  each 
pair  of  consecutive  numbers  in  the  second  column  ; 
then  each  number  in  the  third  column  is  divided  by 
the  one  just  below  it  in  the  second  column,  to  find 
what  per  cent  must  be  added  to  that  number  in 
order  to  obtain  the  one  above  it ;  these  per  cents  are 
arranged  in  a  fourth  column  and  averaged  ;  then  each 
number  in  the  third  column  is  divided  by  the  number 
in  the  second  column  just  above  it,  to  find  what  per 
cent  must  be  subtracted  from  that  number  to  obtain 


f  93  (2).]  CORRECTIONS.  175 

the  number  just  below  it ;  the  per  cents  to  be  sub- 
tracted are  then  arranged  in  a  fifth  column  and 
averaged.  We  may  now  extend  the  second  column 
upwards  by  adding  to  the  first  number  in  it  the  aver- 
age per  cent  from  the  fourth  column,  and  we  may 
extend  it  downward  by  subtracting  from  the  last 
number  the  average  per  cent  found  in  the  fifth 
column.  When  the  second  column  has  been  thus 
extended,  the  corresponding  numbers  in  the  first 
column  may  be  found  by  adding  in  the  temperature 
of  the  room.  The  temperature  at  a  time  which 
would  come  between  the  observations  in  the  series 
thus  extended  may  evidently  be  found  by  simple 
interpolation. 

For  example,  when  the  temperature  of  the  room 
is  26°,  the  observations  below  would  be  reduced  as 
follows :  — 

Temperatures       Temperatures  Fall  of  Per  Cent          Per  Cent  to 

Observed.  less  26°.  Temperature.       to  be  Added,     be  Subtracted. 

40°  .0 


38°.0 


2o.O  5.3  5.0 


;:i     4i     « 

60°.5  340.5  .  ,  .o 

590.0  330.0 

% 

56°.0  30°  .0 

Average 4.9  4.7 

To  extend  the  second  column  upwards  we  add  to 
the  first  number  in  it  4.9  per  cent  of  itself.  Since 
4.9  of  40°.0  is  2°.0,  the  number  above  40°.0  should  be 
40o.O  +  2°.0,  or  42°.0  ;  and  since  4.9  per  cent  of  42°.0 
is  2°.l,  the  next  number  should  be  44°.l,  etc. 

To  extend  the  second  column  downwards,  we  sub- 


176  CALORIMETRY.  [Exp.  32. 

tract  from  the  last  number  (30°. 0)  in  it  not  4.9  per 
cent  but  4.7  per  cent  of  30°. 0  ;  that  is  1°.4  ;  this  gives 
28°.6 ;  and  subtracting  from  this  4.7  per  cent  of  itself, 
or  1°.3,  we  find  27°.3  for  the  number  following,  etc. 

Adding  26°  to  the  new  numbers  in  the  second 
column,  we  infer,  finally,  that  the  temperatures  pre- 
ceding 66°.0  in  the  first  column  should  be  68°.0  and 
70°.l,  while  those  following  56°.0  should  be  54°.6 
and  53°.3,  etc. 

Let  us  suppose  that  the  temperatures  were  ob- 
served at  intervals  of  one  minute;  then  to  represent 
the  temperature  for  instance  1.5  minutes  before  the 
first  recorded  observation,  we  should  take  a  number 
half-way  between  68°.0  and  70°.l,  or  69°.0  nearly. 
If,  however,  the  intervals  between  observations  were 
two  minutes  each,  then  1.5  minutes  would  be  three 
fourths  of  one  interval,  and  we  should  add  to  66° 
three  fourths  of  the  difference  (2°)  between  it  and 
the  next  temperature  above  it  in  the  series  to  find 
the  temperature  (67°.5)  in  question. 

The  discovery  of  various  methods  by  which  the 
calculations  described  above  may  be  shortened,  es- 
pecially by  the  use  of  logarithms,  may  be  left  to  the 
ingenuity  of  the  student.  The  method  here  described 
is  important,  as  an  illustration  of  the  fact  that  when 
a  body  is  steadily  cooling  its  temperature  falls,  not  a 
given  amount  in  each  minute,  but  a  certain  per  cent 
(approximately)  of  the  number  of  degrees  which  lie 
between  it  and  the  temperature  of  the  room  (see 
IF  86,  1). 

The  accuracy  with  which  a  series  of  observations 


1  93  (4).]  CORRECTIONS.  177 

may  be  extended  by  analytical  methods  evidently 
grows  less  as  the  number  of  new  terms  increases.  It 
may  be  said  in  general  that  the  new  terms  should 
not  be  more  numerous  than  those  obtained  by  actual 
observation. 

(3)  HEAT  LOST   BY   COOLING.  —  We  must  distin- 
guish between  the  rate  of  cooling  of  a  calorimeter 
and  the   number  of   units  of  heat  lost  by  it.     The 
latter  may  be  found  without  knowing  the  nature  of 
the  mixture  which  the  calorimeter  contains,  provided 
that  the  inner  cup  is  completely  filled  by  the  mixture, 
or  filled  to  a  known  depth  ;  for  we  have  only  to  refer 
to  the  results  already  found  with  water  at  the  same 
depth  in  Experiment  31. 

If,  for  example,  a  calorimeter,  nearly  filled  with  a 
mixture  of  lead  shot  and  water,  has  been  cooling  for 
ten  minutes  at  an  average  temperature  about  20° 
above  that  of  the  room,  we  reason  that  since  at  a 
temperature  1°  above  that  of  the  room  it  was  found 
(Tf  87)  to  lose  1.8  units  of  heat  per  minute,  at  a 
temperature  20°  above  that  of  the  room  it  would  lose 
20  times  1.8,  or  36  units  per  minute;  that  is,  360 
units  in  ten  minutes.  If,  therefore,  the  first  accurate 
observation  of  temperature  was  taken  ten  minutes 
after  the  introduction  of  the  mixture,  we  should  add 
360  units  to  the  amount  of  heat  apparently  given  out 
by  the  hot  body,  or  if  more  convenient  we  may  sub- 
tract 360  units  from  the  quantity  of  heat  apparently 
absorbed  by  the  cool  body  (see  ^[  98). 

(4)  METHOD  OP  MULTIPLICATION.  —  When  two  ex- 
periments are  made,  in  one  of  which  a  body  is  exposed, 

12 


178  CALORIMETRY.  [Exp.  33. 

let  us  say,  11  times  as  long  or  11  times  as  often  to 
the  air  as  in  the  other  experiment,  in  which  we  give 
it  the  ordinary  exposure,  the  difference  between  the 
results  obtained  in  the  two  cases  should  correspond 
to  the  effect  of  11  less  1,  or  10  ordinary  exposures. 
Hence,  if  this  difference  be  divided  by  10,  we  may 
estimate  roughly  the  correction  to  be  applied  to  the 
result  obtained  with  the  ordinary  exposure. 

If,  for  example,  the  thermal  capacity  of  a  calori- 
meter is  found  to  be  10.1  units  when  warm  water  is 
poured  into  it  directly,  and  11.1  units  if  the  water  is 
first  poured  back  and  forth  five  times  each  way,  then 
the  effect  of  cooling  due  to  10  transfers  is  11.1-10.1, 
or  1  unit  in  the  result ;  and  the  effect  of  a  single 
transfer  is  about  0.1  unit.  The  true  thermal  capacity 
is,  therefore,  about  10.1-0.1,  or  10.0  units.  If  the 
cooling  due  to  transferring  a  substance  from  one 
place  to  another  is  thought  to  affect  the  figure  in  the 
tenths'  place,  as  in  the  example,  it  is  evident  that  the 
hundredths  will  not  be  significant  (see  §  55). 


EXPERIMENT  XXXIII. 

SPECIFIC   HEAT   OF   SOLIDS. 

Tf  94.  Determination  of  the  Specific  Heat  of  a  Solid 
by  the  Method  of  Mixture.  —  I.  A  quantity  of  lead 
shot  sufficient  to  half  fill  the  calorimeter  (Fig.  70, 
^[  85)  is  first  weighed,  then  put  into  a  steam  heater 
(Fig.  79),  and  covered  by  a  cork.  A  thermometer, 
passing  through  the  cork  into  the  midst  of  the  shot, 


IT  94,  L] 


METHOD  OF  MIXTURE. 


179 


FIG.  79. 


is  allowed  to  remain  there  until  it  ceases  to  rise. 
Meanwhile  the  temperature  within  the  calorimeter 
is  determined  by  a  second 
thermometer  (^[  92,  2).  The 
calorimeter  is  then  weighed, 
and  a  vessel  containing  a 
mixture  of  ice  and  water  is 
also  weighed.  This  vessel 
should  be  provided  with  a 
strainer,  so  that  water  may 
be  poured  from  it  without 
danger  of  particles  of  ice  fol- 
lowing the  stream.  The  ice  and  water  should  be 
thoroughly  stirred  just  before  the  experiment,  to 
secure  a  uniform  temperature  of  0°.  The  time  should 
now  be  noted  (^[  92,  9). 

The  thermometer  and  corks  are  then  removed  from 
the  heater,  and  the  shot  is  poured  as  rapidly  as  pos- 
sible (^[  92,  4)  into  the  calorimeter.  Immediately 
ice-cold  water  is  added,  —  the  quantity  being  nearly 
sufficient  to  fill  the  calorimeter.  A  thermometer  is 
then  pushed  cautiously  into  the  middle  of  the  shot 
through  a  small  stopper,  closing  the  inner  cup  water- 
tight (^  92,  8).  The  large  cork  cover  (Fig.  71, 
^[  85)  may  then  be  added,  and  the  time  again  re- 
corded. The  mixture  must  now  be  carefully  shaken. 
The  temperature  indicated  by  the  thermometer  is  to 
be  noted  at  intervals  of  1  minute,  until  it  begins  to 
fall  steadily  (^[  92,  8  and  10).  Then  the  calorimeter 
is  re-weighed  with  its  contents ;  and  the  vessel  origi- 
nally containing  the  water  is  also  weighed  (^[  92,  5). 


180  CALORIMETRY.  [Exp.  33. 

II.  Instead  of  finding  the  temperature  of  the  shot 
in  the  heater,  as  in  I.,  we  may  determine  it  by  a 
series  of  observations  in  the  calorimeter,  before  the 
ice-water  is  added  (^[  92,  10).      It  is  necessary  in 
this  case  to  cork  the  inner  cup,  and  to  shake  the  shot 
between  the  observations  of  temperature  (^[  92,  8), 
in  order  that  there  may  be  a  uniform  temperature 
not  only  in  the  shot,  but  also  in  the  inner  cup,  the 
thermal  capacity  of  which  must  be  considered.     The 
ice-water  is   finally  added,  and  the  temperature  of 
the  mixture  determined  as  before. 

III.  Instead  of  pouring  the  hot  shot  first  into  the 
calorimeter,  we  may  begin  by  introducing  ice-water. 
In  this  case  the  proper  quantity  of  water  must  be 
determined  beforehand.     It  will  probably  be  found 
that  the  water  should  fill  the  calorimeter  about  half- 
full.     In  other  respects  this  method  is  the  same  as  I. 

IV.  Instead  of  assuming  that  the  temperature  of 
the  water  is  the  same  as  that  within  the  vessel  origi- 
nally containing   it    (that   is,  0°),  we    may  find   its 
temperature  after  it  has  been  transferred  to  the  cal- 
orimeter.    In  this  method,  however,  as  in  the  second 
method,  the  thermal  capacity  of  the  cup  must  be  con- 
sidered.   To  avoid  the  necessity  of  making  a  separate 
series  of  observations  (^[  92,  10)  between  which  the 
water  in  the  calorimeter  must  be  shaken  up  (^[  92,  8), 
it  is  customary  to  use  water  at  the  temperature  of 
the  room.     In  this  case,  the  mixture  will  be  above 
the  temperature  of  the  room ;  hence  its  rate  of  cool- 
ing must  be  allowed  for  (^[  93). 

V.  Other  methods   of  determining   specific   heat 


IT  94.]  SPECIFIC  HEAT  OF  SOLIDS.  181 

may  easily  be  devised,  depending  upon  the  use  of 
hot  water  and  cold  shot.  We  have  in  fact  already 
made  use  of  such  a  method  in  finding  the  thermal 
capacity  of  a  calorimeter  (^f  90,  I.).  On  account, 
however,  of  the  practical  difficulties  arising  from 
evaporation  (^[  92,  6),  the  high  temperature  of  the 
^mixture  (*[f  92,  7),  and  the  small  change  of  tempera- 
ture produced,  these  methods  are  generally  avoided. 
The  principal  use  which  can  be  made  of  them  is  as 
a  check  (§45)  upon  results  obtained  in  the  ordinary 
manner. 

The  student  may  observe  that  in  the  second  method 
the  shot  falls  suddenly  in  temperature,  on  account  of 
the  heat  which  it  gives  up  to  the  calorimeter.  This 
heat  is  subsequently  restored  to  the  mixture  when 
the  calorimeter  is  cooled  to  its  original  temperature ; 
hence  in  the  first  method  no  account  need  be  taken 
of  the  thermal  capacity  of  the  calorimeter.  Again 
in  the  fourth  method,  the  cold  water  may  at  first  rise 
rapidly  in  temperature  on  account  of  the  heat  im- 
parted to  it  from  the  calorimeter,  but  this  heat  is 
restored  to  the  calorimeter  when  it  is  again  raised 
by  the  mixture  to  its  original  temperature ;  hence 
in  the  third  method  no  account  need  be  taken  of  the 
thermal  capacity  of  the  calorimeter. 

Instead  of  lead  shot,  copper  or  iron  rivets  may  be 
employed  with  very  slight  modifications  of  the  experi- 
ment. In  the  case  however  of  solids  which  are  soluble 
in  water,  we  must  substitute  for  water  some  other 
liquid  of  known  specific  heat  in  which  the  solids  are 
insoluble  (^[  92,  1).  The  student  may  be  guided  in 


182  CALORIMETRY.  [Exp.  33. 

his  choice  of  methods  by  obvious  considerations  of 
practical  convenience  as  well  as  by  the  principles 
explained  below  in  ^[  95;  but  he  should  make  at 
least  one  determination  of  specific  heat  of  a  solid  by 
the  method  of  mixture  and  reduce  it  as  will  be  ex- 
plained in  ^[  98. 

^[  95.  Comparison  of  Methods  for  the  Determination 
of  Specific  Heat.  —  The  principal  difficulty  in  the  first 
method  (^[  94,  I.),  for  the  determination  of  specific 
heat,  is  to  avoid  a  great  loss  of  heat  while  the  shot  is 
being  transferred  from  the  heater  to  the  calorimeter. 

In  the  second  method  (^[  94,  II.)  there  is  no 
opportunity  for  a  loss  of  heat  on  the  part  of  the  shot, 
since  its  temperature  is  determined  by  a  series  of 
observations  within  the  calorimeter,  from  which  its 
temperature  at  any  point  of  time  may  be  found 
(^[  93,  2).  The  principal  objection  to  the  second 
method  is  the  difficulty  of  determining  accurately  a 
series  of  temperatures  in  which  rapid  changes  take 
place  ;  and  the  necessity  of  allowing  for  the  thermal 
capacity  of  the  calorimeter,  which  is  always  a  more 
or  less  uncertain  quantity,  and  bears  a  considerable 
proportion  to  the  thermal  capacity  of  the  shot. 

The  third  method  (^[  94,  III.)  has  the  same  prac- 
tical advantages  and  disadvantages  as  the  first. 

The  fourth  method  (^[  94,  IV.)  is  the  one  com- 
monly employed  for  the  determination  of  specific 
heat.  Since  the  temperature  of  the  water  is  found 
when  within  the  calorimeter,  there  is  no  opportunity 
(as  in  the  other  methods)  for  heat  to  be  imparted  to 
it  in  the  act  of  pouring.  There  is  however  difficulty, 


1  95.]  SPECIFIC   HEAT  OF  SOLIDS.  183 

as  in  the  second  method,  in  determining  accurately  a 
temperature  which  is  changing  (^[  92,  3),  and  still 
further  difficulty  in  maintaining  a  uniform  tempera- 
ture throughout  the  calorimeter  with  a  quantity  of 
water  which  only  half  fills  it  <^[  92,  8).  When  the 
latter  difficulty  is  avoided  by  using  water  at  the  tem- 
perature of  the  room,  the  mixture  must  have  a  tem- 
perature considerably  above  that  of  the  room,  and 
one  therefore  which  is  hard  to  determine  (^[  92,  3). 
The  thermal  capacity  of  the  calorimeter  must  also, 
as  in  the  second  method,  be  taken  into  account. 

By  comparing  the  results  of  the  first  and  second 
methods,  we  are  able  to  estimate  the  effect  of  the 
heat  lost  in  pouring  the  shot  into  the  calorimeter  (see 
also  ^[  93,  4),  and  by  comparing  results  of  the  third 
and  fourtji  methods,  we  are  able  to  estimate  the  effect 
of  the  heat  absorbed  by  the  ice-cold  water  when  it  is 
poured  from  one  vessel  to  another.  This  will  be 
found  to  be  small  in  comparison  with  the  heat  lost 
by  shot  at  100°  under  similar  circumstances.  The 
second  method,  in  which  the  latter  is  eliminated,  is 
therefore  preferable  to  the  fourth.  In  the  first  and 
third  methods,  the  heat  lost  by  the  shot  is  partly 
offset  by  that  imparted  to  the  water.  Since  the 
former  is  greater  than  the  latter,  the  third  method  is 
preferable  to  the  first;  because  the  longer  exposure 
of  the  water  may  compensate  for  the  more  rapid  cool- 
ing of  the  shot.  The  choice  between  the  second  and 
third  methods  will  depend  largely  upon  the  compara- 
tive accuracy  with  which  we  can  determine  the  heat 
given  out  by  the  calorimeter  (^[  87)  and  the  heat  lost 


184  CALORIMETRY.  [Exp.  34. 

by  the  shot  flf  93,  4).  The  advantages  of  using  in 
any  case  hot  shot  and  cold  water  have  been  already 
stated  flf  94,  V.). 


EXPERIMENT   XXXIV. 

SPECIFIC    HEAT    OF   LIQUIDS. 

^f  96.  Determination  of  the  Specific  Heat  of  a  Liquid 
by  the  Method  of  Mixture.  —  The  specific  heat  of  a 
liquid  may  be  determined  either  by  mixing  it  me- 
chanically with  water,  or  by  bringing  it  in  contact 
with  a  solid  of  known  specific  heat.  The  first  method 
is  the  more  direct,  but  cannot  be  employed  with 
liquids  which  unite  chemically  with  water,  unless 
we  know  the  amount  of  heat  given  out  or  absorbed 
by  the  reaction  (see  ^f  92,  1).  Before  deciding  which 
method  we  shall  employ,  we  therefore  mix  together 
the  contents  of  two  test-tubes,  each  at  the  temper- 
ature of  the  room,  one  containing  water,  the  other 
the  liquid  in  question.  If  no  change  of  temperature 
is  observed,  the  first  method  is  adopted.  If  the  tem- 
perature rises  or  falls,  we  must  either  make  a  sepa- 
rate experiment  to  determine  accurately  the  amount 
of  this  rise  or  fall  (see  Exp.  35),  or  else  adopt  the 
indirect  method,  using  a  solid  instead  of  water. 

I.  The  determination  of  the  specific  heat  of  an 
insoluble  liquid  by  the  method  of  mixture  does  not 
differ  essentially  from  the  case  of  a  solid.  A  heavy 
oil  may  for  instance  be  heated  by  the  same  apparatus 
(Fig.  79,  Tf  94)  employed  for  the  shot,  and  mixed  with 


1T96.]  SPECIFIC   HEAT   OF  LIQUIDS.  185 

ice-cold  water,  according  to  either  of  the  methods 
described  (^[  94).  Instead  of  shaking  the  mixture, 
a  brass  fan  or  stirrer  (Fig.  50,  ^[  65)  may  be  em- 
ployed. 

The  objections  to  mixing  hot  water  with  a  cold 
liquid  are  not  nearly  as  strong  as  in  the  case  of  solids 
(^[  94,  V.)  ;  for  though  most  liquids  have  a  specific 
heat  less  than  that  of  water,  the  differences  are  very 
much  less.  By  pouring  a  comparatively  small  quan- 
tity of  water  at  a  temperature  not  exceeding  40°  or 
50°  into  a  liquid  at  0°  a  mixture  may  be  had  not 
far  from  the  temperature  of  the  room.  With  liquids 
less  dense  than  water  this  method  is  generally  to  be 
preferred  (see  ^[  92,  6  and  8).  The  results  may  be 
reduced  by  the  appropriate  formula  from  ^[  98. 

Attention  has  already  been  drawn  (^[  92,  1)  to 
precautions  against  chemical  action  in  the  case  of 
corrosive  liquids,  and  in  the  case  of  volatile  liquids 
against  evaporation  fl[  92,  6)  and  combustion  («[[  83). 

II.  In  the  case  of  liquids  which  mix  with  water, 
the  ordinary  methods  of  mixture  cannot  generally  be 
employed,  on  account  of  the  heat  absorbed  or  devel- 
oped by  solution  or  combination.  It  is  necessary 
to  find  some  substance,  of  known  specific  heat,  upon 
which  such  a  liquid  exerts  no  thermal  action.  This 
substance  is  then  mixed  with  the  liquid  by  either  of 
the  methods  of  ^  94.  The  data  necessary  for  find- 
ing the  specific  heat  of  the  liquid  are  as  usual  the 
weight  of  the  two  substances  in  question,  the  tem- 
perature of  each  before  the  experiment,  and  the 
resulting  temperature  of  the  mixture. 


186  CALORIMETRY.  [Exp.  34. 

The  lead  shot  already  employed  (^[  94)  may  be 
used  to  determine  m  this  way  the  specific  heat  of 
alcohol,  glycerine,  saline  solutions,  etc.  For  corro- 
sive liquids,  like  nitric  acid,  glass  beads  (of  specific 
heat  about  0.19)  may  be  similarly  employed  (see 
general  formula,  *f[  98).  Evidently  this  indirect 
method  is  more  general  than  the  ordinary  method 
of  mixture,  since  it  can  be  applied  to  all  liquids, 
whether  soluble  or  insoluble  in  water.  It  has  the 
advantage  of  eliminating  almost  completely  the  heat 
lost  by  the  hot  body  between  the  heater  and  the 
calorimeter,  since  this  loss  is  practically  the  same  in 
the  case  of  water  as  in  the  case  of  other  liquids  with 
which  a  comparison  is  made. 

^[  97.  Peculiar  Devices  employed  in  Calorimetry.  — 
In  the  method  of  mixture  (Exps.  33  and  34)  a  thermal 
equilibrium  between  two  or  more  substances  is  estab- 
lished by  bringing  them  in  contact.  It  is  not,  how- 
ever, necessary  that  the  two  bodies  should  touch  each 
other.  The  difficulties  which  arise  from  the  mu- 
tual action  of  two  substances  may  often  be  avoided 
by  surrounding  one  of  them  with  an  envelope,  through 
which,  by  the  conduction  of  heat,  an  equalization  of 
temperature  takes  place.  If,  for  instance,  a  hot  liquid 
contained  in  a  glass  bulb  be  surrounded  by  cold 
water,  a  certain  quantity  of  heat  will  be  given  out. 
Having  found  by  a  separate  experiment  how  much 
heat  is  derived  from  the  bulb  alone,  we  may  calculate 
the  specific  heat  of  the  liquid  in  the  ordinary  manner, 
that  is,  from  the  weights  and  changes  of  temperature 
involved  (see  general  formula,  ^[  98). 


1T97.]  SPECIFIC  HEAT  OF  LIQUIDS.  187 

The  liquid  in  question  may  be  contained  in  an 
ordinary  thermometer  bulb.  In  this  case  its  change 
of  temperature  may  be  inferred  very  accurately  from 
its  contraction,  as  shown  by  the  fall  of  a  column  of 
liquid  in  the  stem  of  the  thermometer.  It  is  neces- 
sary, of  course,  to  make  a  careful  comparison  of  a 
thermometer  containing  an  unknown  liquid  with  an 
ordinary  mercurial  thermometer  (see  ^f  92,  2).  This 
method  has  obvious  advantages  in  the  case  of  costly 
liquids. 

On  the  other  hand,  when  the  supply  of  a  fluid  is 
unlimited,  it  is  frequently  advantageous  to  use  an 
envelope  in  the  form  of  a  spiral  tube,  or  coil,  through 
which  the  fluid  in  question  may  be  passed  in  a  con- 
tinuous stream.  We  are  thus  enabled  to  bring  a 
great  volume  of  the  fluid  in  thermal  equilibrium  with 
a  small  volume  of  water.  This  device  is  exceedingly 
important  in  the  case  of  gases,  since  it  would  be 
otherwise  impossible  to  bring  enough  gas  in  thermal 
equilibrium  with  a  given  quantity  of  water  to  af- 
fect the  temperature  of  the  water  by  a  measurable 
amount. 

The  weight  of  the  gas  employed  is  not  measured 
directly,  but  is  determined  from  its  density  (see 
Tf^T  44,  46)  and  from  the  volume  employed.  The 
volume  is  indicated  by  a  gas-meter  (aJ,  Fig.  80) 
through  which  the  gas  is  first  passed.  The  gas  is 
then  raised  to  the  temperature  of  100°  by  passing 
it  through  a  steam  jacket,  Id.  Then  it  circulates 
through  a  coiled  tube  surrounded  with  water,  and 
escapes  from  an  orifice  where  its  final  temperature 


188  CALORIMETRY.  [Exp.  34. 

can  be  observed.  From  the  thermal  capacity  and 
rise  of  temperature  of  the  calorimeter,  we  may  cal- 
culate the  quantity  of  heat  given  out  by  a  known 
quantity  of  gas  in  falling  through  a  known  number 
of  degrees,  and  hence  the  specific  heat  of  the  gas. 
It  is  found  that  the  specific  heat  of  air  at  the  con- 
stant pressure  of  one  atmosphere  is  about  0.238,  or  a 


FIG.  80. 

little  less  than  one  fourth  that  of  an  equal  weight  of 
water. 

A  much  more  difficult  task  consists  in  the  determi- 
nation of  the  specific  heat  of  a  gas  when  confined  to 
a  constant  volume.  The  following  method  is  sug- 
gested. It  depends  upon  the  fact  that  a  given  electric 
current  passing  for  a  given  time  through  a  given  con- 
ductor generates  in  that  conductor  a  given  quantity 
of  heat.  This  quantity  may  be  found  by  experiment 
(see  Exp.  86),  or  calculated  by  the  principles  of 
§  136.  Let  us  suppose  that  a  known  quantity  of  heat 
is  thus  suddenly  generated  within  a  closed  flask  (Fig. 
81)  ;  and  that  the  increased  pressure  of  the  air  is 


IT  97.]  SPECIFIC   HEAT  OF   GASES.  189 

measured,  as  in  If  80,  by  the  rise  of  mercury  in  an 
open  tube.  Then  the  average  temperature  of  the 
air  within  the  flask  can  be  calculated  (see  §  76). 

We  may  therefore  find  the  ther- 
mal capacity  of  a  known  volume  or 
of  a  known  weight,  and  hence  the 
specific  heat  in  question  (about  .169). 

It  is  found  that  the  thermal  ca- 
pacity of  a  cubic  metre  of  air  is 
about  219  units  at  0°  and  76  cm.  when 
prevented  from  expanding,  as  against 
308  units  when  free  to  expand  under 
a  constant  pressure.  The  thermal 
capacity  of  an  equal  volume  of  oxy- 
gen, of  nitrogen,  or  of  hydrogen  is  very  nearly  the 
same  as  that  of  air  under  similar  conditions. 

Instead  of  using  an  electrical  current  to  generate 
heat  (as  illustrated  in  Fig.  81),  we  may  employ  vari- 
ous other  agents,  as  for  instance  the  combustion,  the 
solidification,  the  fusion,  the  condensation,  or  the 
vaporization  of  a  known  weight  of  a  given  substance, 
or  the  conversion  through  friction  of  a  given  amount 
of  work  into  heat  (see  Exp.  70).  If,  for  example, 
the  combustion  of  a  gram  of  coal  heats  a  kilogram  of 
water  8°,  and  a  kilogram  of  petroleum  16°  ;  or  if  100 
grams  of  ice  cool  these  liquids  8°  and  16°  respec- 
tively;  the  specific  heats  must  be  to  each  other  as 
2  to  1.  The  same  inference  would  be  drawn  if  the 
same  quantity  (100  grams)  of  steam  which  heats 
1  kilogram  of  water  54°  were  found  to  heat  2  kilo- 
grams of  petroleum  by  the  same  amount.  The  spe- 


190  CALOEIMETRY.  [Exp.  34. 

cific  heats  of  different  substances  are  to  each  other,  in 
general,  inversely  as  the  changes  of  temperature  pro- 
duced by  a  given  cause,  and  also  inversely  as  the 
weights  affected.  The  determination  of  specific  heat 
is  evidently  capable  of  as  many  modifications  as  there 
are  different  methods  by  which  a  definite  quantity  of 
heat  may  be  generated  or  absorbed. 

Instead  of  using  the  pressure  of  air  to  measure  its 
temperature,  we  may  also  employ  its  expansion  (§  80) 
as  in  the  air  thermometer  (If  74).  The  specific  heat 
of  air  under  a  constant  pressure  might  obviously  be 
determined  by  an  apparatus  similar  to  that  repre- 
sented in  Fig.  81 ;  hence,  conversely,  if  this  specific 
heat  is  known,  we  may  measure  quantities  of  heat  by 
the  expansion  which  they  produce  in  air  at  a  given 
pressure.  It  does  not  (as  one  might  think)  make 
any  difference  theoretically  hoiv  much  air  is  heated ; 
because  an  increase  in  the  quantity  of  air  will  be 
offset  by  a  decrease  in  the  temperature  to  which  it 
will  be  raised  by  a  given  amount  of  heat ;  and  for  the 
same  reason  it  is  indifferent  whether  a  small  portion 
of  the  air  is  heated  a  great  deal,  or  whether  a  con- 
siderable portion  is  heated  by  a  proportionately  small 
amount.  In  this  method  of  estimating  heat  it  is  not 
necessary  to  wait  for  an  equilibrium  of  temperature. 
We  hasten  in  fact  to  make  our  observations  before  an 
equilibrium  is  reached,  so  as  to  avoid  loss  of  heat  by 
contact  of  the  air  with  the  sides  of  the  vessel  in  which 
it  is  contained.  It  has  been  calculated  that  one  unit 
of  heat  should  in  all  cases  cause  in  a  body  of  air  at 
76  cm.  pressure  an  expansion  of  about  12  cubic  centi- 


IT  97.]  ICE   CALORIMETER.  191 

metres.  Since,  an  expansion  of  less  than  1  cubic  milli- 
metre is  easily  detected,  we  have,  in  the  air  ther- 
mometer, a  very  delicate  means  of  measuring  small 
quantities  of  heat.1 

Instead  of  air,  we  may  use  any  other  fluid  which 
has  a  regular  rate  of  expansion  to  determine  quanti- 
ties of  heat.  The  principle  above  explained  has 
been  applied  by  Favre  and  Silbermann  in  the  con- 
struction of  their  mercury  calorimeter.2  This  is  essen- 
tially a  thermometer  with  a  huge  bulb.  If  even 
a  small  quantity  of  hot  liquid  be  introduced  into  a 
cavity  in  this  bulb,  there  will  be  a  perceptible  expan- 
sion of  the  mercury,  by  which  we  may  measure 
the  heat  given  out  by  the  liquid  in  question  ;  for  it 
has  been  found  that  1  unit  of  heat  always  causes  in 
a  body  of  mercury  an  expansion  of  about  4  cubic 
millimetres. 

There  are  various  other  definite  effects  produced 
by  a  given  quantity  of  heat,  any  one  of  which  might 
theoretically  be  applied  to  the  pur- 
poses of  calorimetry.  The  only  ap- 
plication of  practical  importance 
depends,  however,  upon  the  heat 
required  for  the  fusion  of  ice  (see 
Experiment  36).  A  rough  form  of 
ice  calorimeter  consists  of  a  block  of  ice  (Fig.  82) 
with  a  small  cavity  in  which  a  hot  body  may  be 

1  The  air  thermometer  has  been  used  in  the  Jefferson  Physical 
Laboratory  to  measure  minute  quantities  of  heat  generated  in  a  car- 
bon fibre  by  telephone  currents. 

2  See  Ganot's  Physics,  §  463. 


192  CALORIMETRY.  [Exr.  34. 

placed.  A  second  block  may  be  used  as  a  cover. 
The  water  formed  by  the  liquefaction  of  ice  is  gathered 
by  a  sponge,  and  weighed  by  the  usual  method  of  dif- 
erence.  Since  one  unit  of  heat  melts  one-eightieth 
of  a  gram  of  ice,  the  quantity  of  heat  given  out  by 
the  body  in  falling  to  a  temperature  of  0°  can  easily 
be  calculated.  In  Bunsen's  ice  calorimeter,  the  quan- 
tity of  ice  melted  is  estimated  by  the  change  in  volume 
of  a  mixture  of  ice  and  water. 

T[  98.  Calculation  of  Specific  Heat  in  the  Method  of 
Mixture.  —  If  wl  is  the  weight  of  the  body,  the  specific 
heat  of  which  («x)  is  to  be  determined,  and  tt  the 
temperature  of  this  body,  reduced  to  the  time  of 
mixing  ;  if  w2  is  the  weight  of  the  body  the  specific 
heat  (&,)  of  which  is  known,  and  if  t2  is  its  tempera- 
ture, also  reduced  to  the  time  of  mixing  ;  if  c  is  the 
thermal  capacity  of  the  calorimeter,  t3  its  original  tem- 
perature and,  t  the  temperature  of  the  mixture  ;  then 
if  q  is  the  quantity  of  heat  lost  by  cooling,  that  is, 
absorbed  by  the  air,  etc.,  we  have,  by  the  principle 
of  §  90,  the  general  formula, 


From  this  formula  we  may  obtain  the  solution  of 
all  problems  in  the  determination  of  specific  heat  by 
the  method  of  mixture. 

In  addition  to  «,,  c,  and  q  (which  are  known,  or 
may  be  calculated),  we  require  at  least  five  data 
for  a  determination  of  specific  heat  ;  namely,  the  two 
weights  employed,  wl  and  «0,,  the  two  corresponding 
temperatures,  ^  and  ^,  also  the  temperature,  £,  of 


IT  98.]  CALCULATIONS  OF   SPECIFIC   HEAT.  193 

the  mixture.  The  original  temperature,  £8,  of  the  ca- 
lorimeter must  also  be  determined,  unless  by  the 
nature  of  the  experiment  it  is  known  to  agree  with 
one  of  the  other  temperatures. 

When  water  is  used  s2  =  1  ;  hence  we  have,  if  the 
water  used  is  colder  than  the  mixture, 


or  if  the  water  is  warmer  than  the  mixture, 


If  the  temperature  of  the  water  is  taken  in  the  ca- 
lorimeter, so  that  £2  =  £8,  we  may  combine  the  terms 
in  the  numerator,  so  that  for  cold  water, 

(w2  +  c}(t—  *8)  +  g 
*i  -  Wl  (^  _  t) 

or  for  hot  water, 


—  0-9. 


TV 


If  the  original  temperature  of  the  calorimeter  is 
the  same  as  that  of  the  mixture,  the  terms  c  (t  —  £3) 
and  c  (£8  —  t)  disappear  from  I.  and  II.  respectively  ; 
hence,  for  cold  water, 


_    t      ~i  .  v 

- 


and  for  hot  water, 


194  CALORIMETRY.  [Exp.  35. 

If,  finally,  the  temperature  of  the  mixture  is  the 
same  as  that  of  the  room,  there  is  no  loss  of  heat  by 
cooling  (§  89),  that  is,  q  =  0  ;  hence  the  term  q 
disappears  from  all  the  formulas.  We  have  therefore 
in  the  simplest  possible  case,  when  the  calorimeter  is 
at  the  temperature  of  the  room  both  before  and  after 
the  experiment,  if  cold  water  is  used, 

s  =  ^t-t2)  yiL 


and  if  hot  water  is  used, 

_W2(t2-f) 


VIII. 


The  calculation  of  the  thermal  capacity  of  the 
calorimeter  (c)  is  explained  in  ^[  86  and  91  ;  that 
of  the  heat  lost  (q)  in  ^  93,  3.  The  correction  of  the 
temperatures  tl  and  t2  to  the  time  of  mixing  may  be 
done  either  by  graphical  or  by  analytical  methods 
(t  93,  1  and  2). 


EXPERIMENT   XXXV. 

HEAT   OF   SOLUTION. 

^[  99.  Determination  of  Latent  Heat  of  Solution.  — 
When  a  solid  dissolves  in  a  liquid,  or  when  two 
liquids  mix  together,  there  is  almost  always  a  rise 
or  fall  of  temperature.  This  is  due  probably  to  a 
a  molecular  re-arrangement  which  takes  place.  The 
object  of  this  experiment  is  to  find  how  much  heat  is 
given  out  or  absorbed,  as  the  case  may  be,  by  one 


«f  99.]       LATENT  HEAT  OF  SOLUTION.        195 

gram  of  a  given  substance  when  mixed  with  or  dis- 
solved in  water. 

I.  LIQUIDS.  —  When  equal  volumes  of  alcohol  and 
water  are  mixed  together  (see  ^[  96)  a  rise  of  tem- 
perature may  be  observed.  To  measure  this  rise 
accurately,  a  calorimeter  is  to  be  weighed  empty, 
and  re-weighed  with  a  quantity  of  alcohol  which  fills 
it  half-full,  and  which  is  at  a  temperature,  accurately 
observed,  not  far  from  that  of  the  room.  An  equal 
volume  of  water,  heated  or  cooled  if  necessary  so  as 
to  have  exactly  the  same  temperature,  is  then  mixed 
with  the  alcohol  in  the  calorimeter,  and  the  resulting 
temperature  accurately  determined  by  a  series  of  ob- 
servations (^[  92,  10).  The  weight  of  water  is  also 
to  be  found  (see  ^[  92,  5).  If  the  thermal  capacity 
of  the  calorimeter  and  the  specific  heat  of  the  liquid 
are  both  known,  the  latent  heat  of  solution  may  be 
calculated  by  formula  II.,  ^[  100. 

It  is  better,  however,  to  repeat  the  experiment  with 
water  at  a  much  lower  temperature,  which  must  be 
determined  (see  «[[  92, 10)  by  a  series  of  observations. 
The  object  aimed  at  is  to  offset  in  this  way  the  heat 
due  to  mixture.  When  alcohol  in  a  calorimeter  at 
the  temperature  of  the  room  is  mixed  with  an  equal 
volume  of  water,  which  is  cooler  than  it  by  the  right 
number  of  degrees,  scarcely  any  rise  or  fall  of  temper- 
ature will  be  observed  in  the  calorimeter.  In  this 
case  a  single  observation  will  suffice. 

Let  us  suppose,  for  example,  that  equal  volumes  of 
alcohol  and  water  rise  8°  when  mixed  at  the  same 
temperature,  but  that  if  the  water  is  9°  cooler  than 


196  CALOKIMETKY.  [Exp.35. 

the  alcohol,  the  rise  is  2°.  Then  since  9°  in  the 
water  makes  a  difference  of  8°  —  2°,  or  6°,  in  the 
mixture,  12°  in  the  water  would  make  a  difference 
of  8°  in  the  mixture.  It  follows  that  the  alcohol 
could  be  mixed  with  an  equal  volume  of  water  12° 
below  it  in  temperature  without  being  warmed  or 
cooled  by  the  process. 

It  would  be  well  to  test  the  accuracy  of  such  a 
conclusion  by  a  third  experiment.  When  the  desired 
difference  of  temperature  has  been  found,  either  by 
experiment  or  by  calculation,  the  latent  heat  of  mix- 
ing is  easily  computed.  We  multiply  the  weight  of 
water  by  its  rise  of  temperature  to  find  the  number 
of  units  of  heat  received,  and  divide  by  the  weight 
of  alcohol  to  find  the  amount  given  out  by  one  gram ; 
or  we  may  use  formula  III.,  ^[  100. 

The  experiment  may  be  varied  by  using  different 
liquids,  or  by  mixing  a  given  liquid  with  water  in 
different  proportions. 

II.  SOLIDS.  —  When  ammonic  nitrate  is  dissolved  in 
water  a  fall  of  temperature  is  observed.  The  amount 
of  this  fall  may  be  determined  as  in  the  case  of  alco- 
hol ;  but  in  order  that  the  solid  may  be  readily  dis- 
solved, it  is  better  to  use  only  one  part  of  the  salt  in 
nine  of  water.  To  ensure  rapid  solution,  the  salt 
should  be  pulverized.  In  the  first  experiment  the  salt, 
the  water,  and  the  calorimeter  should  all  start  at  the 
temperature  of  the  room.  The  fall  of  temperature 
of  the  water  may  require  a  thermometer  divided  yito 
tenths  of  degrees  for  its  accurate  determination.  The 
use  of  a  stirrer  is  very  important  (^f  65,  5). 


f  100.]  LATENT  HEAT  OF  SOLUTION.  197 

The  experiment  may  now  be  repeated  with  water 
somewhat  warmer  than  before,  with  a  view  to  mak- 
ing the  resulting  temperature  agree  with  that  of  the 
room.  The  water  should,  however,  be  placed  first  in 
the  calorimeter,  in  order  that  the  temperature  of  the 
latter  may  be  accurately  determined.  A  series  of 
observations  must  be  taken  (^[  92,  10).  The  salt  is 
finally  added,  and  the  fall  of  temperature  accurately 
measured.  If  the  water  has  been  heated  too  much 
or  too  little,  the  experiment  may  be  repeated  until 
the  mixture  agrees  in  temperature  with  the  room ;  or 
the  desired  temperature  of  the  water  may  be  calcu- 
lated by  the  same  process  of  reasoning  as  was  em- 
ployed in  I.  In  calculating  the  latent  heat  of  solution 
by  this  method,  the  thermal  capacity  of  the  calori- 
meter must  be  taken  into  account,  since  part  of  the 
heat  absorbed  by  the  salt  is  supplied  by  the  calori- 
meter. In  other  respects  the  reduction  is  the  same 
as  in  I.  (see  also  formula  IV.,  ^[  100). 

If,  for  instance,  10  grams  of  salt  cool  90  grams  of 
water  contained  in  a  calorimeter  with  a  thermal  ca- 
pacity equal  to  10  units,  from  22°  to  20°,  that  is  2°, 
we  have  (90  -j-  10)  X  2  =  200  units  of  heat  given  out. 
Since  10  grams  of  the  salt  absorb  200  units,  each 
gram  must  require  20  units  of  heat ;  hence  the  latent 
heat  of  solution  is  20.  The  latent  heat  in  question 
varies  slightly  according  to  the  strength  of  the  solu- 
tion formed. 

^f  100.  Calculation  of  the  Latent  Heat  of  Solution. 
—  If  Wi  is  the  weight  of  the  substance  whose  latent 
heat  of  solution,  l»  is  to  be  determined,  *x  its  specific 


198  CALORIMETKY.  [Exp.  35. 

heat,  and  tv  its  original  temperature;  if  w2  is  the 
weight  of  the  solvent,  s2  its  specific  heat,  and  t2  its 
original  temperature  ;  if  c  is  the  thermal  capacity,  ts 
the  original  and  t  the  final  temperature  of  the  calori- 
meter (hence  also  of  the  mixture),  then  the  quan- 
tities of  heat  absorbed  are,  (1)  wv  st  (t  —  ^)  in 
raising  the  temperature  of  the  substance  dissolved  ; 
(2)  w2  s2  (t  —  £2)  in  raising  the  temperature  of  the 
solvent  ;  and  (3)  c  (t  —  ts)  in  raising  the  tempera- 
ture of  the  calorimeter  and  (4)  wl  ^  in  the  act  of 
solution.  Hence,  by  the  principle  of  §  90, 


neglecting  the  heat  lost  by  cooling. 

This  gives  for  the  latent  heat  of  mixing  with  water, 
which  we  consider  positive  if  heat  is  absorbed,  but 
negative  if  (as  is  usually  the  case  when  two  liquids 
are  mixed)  heat  is  given  out,1  since  s2  =  1,  and  since 
ti  and  ts  are  the  same  (the  temperature  of  the  liquid 
being  determined  in  the  calorimeter), 


If  the  experiment  is  varied  so  that  t  =  t1  then  we 
have  simply 

_^(*-*Q  nL 

Wl 

If,  however,  the  temperature  of  the  water  is  found 
within  the  calorimeter,  so  that  t2  =  tm  the  substance 

1  The  same  formula  may  be  used  to  determine  the  heat  of  combi- 
nation, only  that  the  sign  must  be  reversed  (see  1T  106). 


T  101.]          LATENT   HEAT  OF  LIQUEFACTION.  199 

dissolved  being  as  before  unchanged  in  temperature, 
we  have  for  the  latent  heat  of  solution,  which  we 
call  positive  when  heat  is  absorbed,  the  formula 


wl 


EXPERIMENT  XXXVI. 

LATENT   HEAT   OF   LIQUEFACTION. 

^J  101.  Determination  of  the  Latent  Heat  of  Water. 
—  Latent  heats  of  liquefaction  are  determined  in 
essentially  the  same  manner  as  latent  heats  of  solu- 
tion (Exp.  35,  II.).  Instead,  however,  of  dissolving 
a  solid  in  a  fluid,  the  solid  is  simply  melted  by 
the  fluid.  Knowing  the  weights,  specific  heats,  and 
changes  of  temperature  of  the  substances  in  question, 
we  may  calculate  by  the  general  formula  (^[  100,  I.) 
the  heat  required  to  melt  one  gram  of  the  solid ;  or, 
in  other  words,  its  latent  heat  of  liquefaction. 

It  is  evident  that  the  liquid  must  exert  no  solvent 
action  on  the  solid,  otherwise  we  should  have  to 
allow  for  heat  of  solution  (see  Exp.  35).  It  is  also 
necessary  that  the  mixture  be  at  a  higher  tempera- 
ture than  the  solid,  else  the  solid  will  not  melt.  It 
is  well  that  the  solid  should  start  at  its  melting-point, 
since  otherwise  we  must  allow  for  the  heat  necessary 
to  raise  it  to  the  temperature  in  question.  A  consid- 
erable time  must  generally  be  allowed  for  the  process 
of  melting ;  to  shorten  this  time  as  much  as  possible, 


200  CALORIMETKY.  [Exp.  36. 

the  mixture  should  be  vigorously  stirred.  Observa- 
tions of  temperature  should  be  taken  from  time  to 
time  (^[  92,  8)  during  the  process. 

When  ice  is  the  solid  employed,  difficulty  will  be 
found  in  obtaining  sufficiently  small  pieces  free  from 
water.  The  ice  should  be  cracked  into  fragments 
weighing  a  few  grams  each,  which  are  then  to  be 
wrapped  up  in  cotton-waste  and  weighed.  Any 
moisture  formed  by  the  melting  of  the  ice  should 
be  absorbed  by  the  waste. 

The  calorimeter  is  weighed  empty,  and  re-weighed 
when  about  half-full  of  warm  water.  The  tempera- 
ture of  the  water  should  be  about  50°,  and  is  deter- 
mined by  a  series  of  observations  (^[  92,  10)  ;  then 
ice  is  added  until  the  calorimeter  is  nearly  full.  The 
ice  should  be  handled  by  means  of  a  portion  of  the 
cotton  waste  which  surrounds  it,  and  each  fragment 
should  be  wiped  as  dry  as  possible  before  placing  it 
in  the  calorimeter.  The  time  occupied  by  this  pro- 
cess and  by  the  fusion  of  the  ice  should  be  noted 
(^[  92,  9).  The  resulting  temperature  of  the  water 
must  be  accurately  determined.  The  quantity  of  ice 
used  should  be  found  both  by  re-weighing  the  cotton 
waste  and  by  re-weighing  the  calorimeter  (^[  92,  5). 

^|  102.  Calculation  of  the  Latent  Heat  of  Water.  — 
If  wl  is  the  weight  of  ice  employed,  ^  its  original 
temperature  (that  is,  0°)  and  sl  its  specific  heat  in  the 
liquid  state  (that  is,  1)  ;  if  w2  is  the  weight  of  water 
employed,  £„  its  temperature  reduced  to  the  time  of 
mixing  (^[  93),  and  s2  its  specific  heat  (that  is  1)  ;  if 
c  is  the  thermal  capacity  of  the  calorimeter  calculated 


If  102.]          LATENT   HEAT  OP  LIQUEFACTION.  201 

as  in  Tf  91,  t3  its  original  temperature  (the  same  as  £2), 
and  t  the  temperature  of  the  mixture  ;  we  have,  sub- 
stituting these  values  in  formula  II.,  ^[  100,  — 


From  the  numerator  of  this  fraction  should  be  sub- 
tracted a  correction  expressing  the  number  of  units 
of  heat  lost  by  the  warm  water  while  the  ice  is  being 
melted.  Since  the  water  begins  at  a  temperature  £2, 
and  ends  at  a  temperature  t,  its  average  temperature 
is  I  (£2  -|-  «),  nearly.  Subtracting  the  temperature  of 
the  room,  we  have,  approximately,  the  average  excess 
of  temperature.  Multiplying  as  in  ^[  93  (3),  by  the 
number  of  minutes  required  to  melt  the  ice,  and  also 
by  the  heat  lost  per  minute  when  the  temperature  is 
1°  above  that  of  the  room  (see  ^[87),  we  have  the 
correction  in  question.  Evidently,  if  the  average 
temperature  of  the  water  is  the  same  as  that  of  the 
room,  no  correction  for  cooling  need  be  made. 

The  truth  of  the  formula  for  the  latent  heat  of 
water  may  be  seen  by  the  following  considerations  : 
Since  wz  grams  of  water  and  the  equivalent  of  c 
grams  of  water  (in  the  brass  and  other  materials 
composing  the  calorimeter)  are  cooled  from  £2°  -to  t°, 
the  heat  lost  by  the  hot  bodies  amounts  to  (wz  -\-  c) 
X  (£2  —  0  units.  Subtracting  from  this  the  correction 
for  cooling,  we  have  a  remainder  which  must  repre- 
sent the  heat  absorbed  by  the  cold  bodies  ;  that  is, 
the  ice  and  the  water  formed  by  its  liquefaction. 
Now  wl  grams  of  ice  form  wl  grams  of  water  at  0°  ; 


202  CALORIMETRY.  [Exp.  37. 

and  to  raise  this  to  t°  requires  wl  X  t  units  of  heat. 
Subtracting  this  from  the  previous  remainder,  we 
have,  therefore,  the  heat  required  to  melt  wl  grams 
of  ice.  Finally,  dividing  by  w^  we  have  the  heat 
required  to  melt  1  gram,  or  the  latent  heat  in 
question. 


EXPERIMENT  XXXVII. 

LATENT    HEAT    OF    VAPORIZATION. 

^[  103.  Determination  of  the  Latent  Heat  of  Steam. 
—  There  are  many  points  of  resemblance  between 
the  determination  of  the  latent  heat  of  vaporization 
and  that  of  the  latent  heat  of  liquefaction  (Exp.  36). 
Instead  of  melting  a  solid  in  a  liquid,  a  vapor  is  con- 
densed in  a  liquid.  From  the  weights,  specific  heats, 
and  changes  of  temperature  in  question,  latent  heats 
of  vaporization  may  be  calculated  by  the  same  general 
formula  (^[  100,  I.)  as  latent  heats  of  liquefaction. 

The  vapor  must  evidently  have  no  chemical  affinity 
for  the  liquid.  The  liquid  must  be  at  lower  tem- 
perature than  the  vapor,  in  order  that  the  latter  may 
be  condensed.  The  vapor  should  start  as  nearly  as 
possible  at  its  temperature  of  condensation,  otherwise 
an  allowance  must  be  made  for  the  heat  given  out 
in  reaching  this  temperature.  Care  must,  however, 
be  taken  that  the  vapor  is  freed  from  particles  of 
liquid  formed  by  its  condensation,  before  it  passes 
into  the  calorimeter. 


IF  103.]  LATENT   HEAT  OF   VAPORIZATION.  203 

When  steam  is  used,  it  is  passed  from  a  generator 
(a,  Fig.  83)  through  a  trap  (6),  where  nearly  all  its 
moisture  is  deposited. 
It  will  be  seen  in  the 
diagram  that  the  exit 
tube  is  completely  sur- 
rounded, either  by  steam 
or  by  cork,  until  it 
reaches  the  calorimeter. 
If,  therefore,  this  tube 
is  well  heated  by  a  cur- 
rent of  steam  before  the 
experiment,  there  is  no 

reason    why   any    condensation    should    take    place 
within  it. 

The  calorimeter  is  weighed  when  empty,  and  re- 
weighed  with  a  quantity  of  water  sufficient  nearly 
to  fill  the  inner  cup,  and  as  cold  as  possible.  The 
temperature  of  this  water  is  determined  by  a  series 
of  observations  at  intervals  of  one  minute  (^[  92, 10) ; 
then  the  current  of  steam  issuing  from  the  trap  is 
turned  suddenly  into  the  water.  The  water  is  stirred 
vigorously  by  twisting  the  stem  of  a  thermometer  to 
which  a  stirrer  is  attached.  When  the  temperature 
of  the  water  has  risen  as  much  above  that  of  the 
room  as  it  was  below  it  before  the  admission  of 
steam,  the  trap  is  taken  away  from  the  calorimeter, 
and  the  resulting  temperature  determined  by  another 
series  of  observations.  The  time  used  in  heating  the 
water  to  the  required  temperature  should  be  as  small 
as  possible,  to  avoid  errors  due  to  gain  or  loss  of 


204  CALOKIMETRY.  [Exp  37 

heat  ;  but  if  the  average  temperature  agrees  with 
that  of  the  room,  no  correction  for  cooling  need  be 
applied  (see  ^[  102).  The  weight  of  steam  con- 
densed is  found  by  re-weighing  the  calorimeter,  and 
the  temperature  of  this  steam  determined  by  an  ob- 
servation of  the  barometer  (see  ^[  69,  II.). 

T[  104.  Calculation  of  the  Latent  Heat  of  Steam.  — 
If  w^  is  the  weight  of  steam  condensed,  *x  the  specific 
heat  of  the  liquid  formed  by  its  condensation  (that  is, 
I),1  and  £j  its  original  temperature  (let  us  say  100°, 
but  see  Table  14)  ;  if  w2  is  the  weight  of  water,  s2  its 
specific  heat  (that  is,  1)  and  tz  its  original  tempera- 
ture ;  if  c  is  the  thermal  capacity  of  the  calorimeter, 
ts  its  original  temperature  (the  same  as  £2),  and  t  the 
temperature  of  the  mixture  ;  we  have,  substituting 
these  values  in  the  general  formula  (^[  100,  I.),  — 

Q2  +  c)   (t-t,)  -w,  (100-Q 


To  the  numerator  of  this  fraction  should  be  added 
the  heat  (if  any)  lost  in  cooling,  since  this  is  also  at 
the  expense  of  the  steam. 

The  formula  may  also  be  established  by  a  process 
of  reasoning  similar  to  that  used  in  ^[  102.  To  raise 
the  equivalent  of  w2  +  c  grams  of  water  (t  —  £2)  de- 
grees requires  (w,  +  c)  X  (t  —  £2)  units  of  heat.  Part 
of  this  was  furnished  by  the  ivl  grams  of  water  at 
100°  (nearly)  in  cooling  to  t°.  This  part  is  clearly 
w^  (100  —  t).  Subtracting  this  from  the  total  heat 

1  The  specific  heat  of  water  varies  from  1.000  at  0°  to  1.013  at 
100°,  having  a  mean  value  of  about  1.005. 


t  105  ]  HEAT  OF  COMBINATION.  205 

received  by  the  water,  we  have  that  given  up  to  it  by 
w^  grams  of  steam  in  the  act  of  condensation ;  hence, 
dividing  by  w19  we  have  the  heat  given  out  by  one 
gram  of  steam  at  100°  when  condensed  into  water  at 
100° ;  that  is,  the  latent  heat  in  question. 


EXPERIMENT   XXXVIII. 

HEAT   OP   COMBINATION. 

^[  105.  Determination  of  Heats  of  Combination.  — 
The  same  method,  essentially,  is  employed  for  the 
determination  of  heats  of  combination  as  for  heats  of 
solution  (Experiment  35)  ;  the  only  difference  being 
that  the  solvent  has  a  chemical  affinity  for  the  sub- 
stance dissolved.  From  the  weights,  specific  heats, 
and  changes  of  temperature  of  the  materials  involved, 
the  heat  of  combination  may  be  calculated  by  the 
general  formula  (1  100,  I.).  Heats  of  combination 
are,  however,  called  positive  when  the  result  of  mix- 
ture is  to  raise  the  temperature  of  the  constituents. 

(1)  ZINC  AND  NITRIC  ACID. — A  gram  of  pure 
zinc  filings  is  to  be  dissolved  in  at  least  fifty  times  its 
weight  of  dilute  nitric  acid.  The  student  should  de- 
termine by  a  preliminary  experiment  what  strength 
of  acid  may  be  required  to  ensure  rapid  solution 
without  danger  of  accident  from  excessive  efferves- 
cence. This  will  depend  largely  upon  the  fineness 
of  the  zinc.  When  "  zinc  dust "  is  used,  very  dilute 
acids  must  be  employed.  The  zinc  dust  should  be 


206  CALORIMETRY.  [Exp.  38. 

poured  into  the  acid,  not  the  acid  on  the  zinc  dust. 
The  inner  cup  of  the  calorimeter  (Fig.  71,  ^[  85) 
should  be  replaced  by  one  of  glass  (^[  92,  1),  the 
thermal  capacity  of  which  must  be  calculated  as  in 
1 91.  The  glass  cup  is  then  nearly  filled  (f  92,  8)  with 
the  dilute  acid  at  a  temperature  below  that  of  the 
room.  This  temperature  must  not,  however,  be  so 
low  as  to  arrest  the  chemical  action.  The  process  of 
solution  may  be  greatly  accelerated  by  the  use  of  a 
platinum-stirrer ; l  but  a  brass  stirrer  coated  with 
asphaltum  may  be  employed  (see  ^[  92,  1).  The 
quantity  of  dilute  acid  used  must  be  found  by  weigh- 
ing the  calorimeter  with  and  without  it ;  and  the 
rise  of  temperature  of  this  acid  must  be  determined 
by  a  series  of  observations  of  temperature  (^[  92,  10) 
both  before  and  after  the  experiment.  It  is  well 
also  to  re-weigh  the  calorimeter  after  the  experiment, 
to  guard  against  any  loss  of  material  (^[  92,  5).  The 
loss  of  weight  due  to  the  escape  of  nitric  oxide  gas 
will  hardly  be  detected. 

(2)  ZINC  OXIDE  AND  NITRIC  ACID.  —  The  experi- 
ment is  now  to  be  repeated  with  a  quantity  of  zinc 
oxide  which  would  be  formed  by  the  combustion  of 
1  gram  of  zinc.  This  quantity  is  1.25  g.,  very  nearly. 
The  same  weight  and  strength  of  acid  are  to  be  used 
as  before  (1)  ;  but  the  temperature  should  be  very 
little  below  that  of  the  room. 


1  Currents  of  electricity  generated  by  the  contact  of  platinum  and 
zinc  assist  the  chemical  action.  It  is,  indeed,  stated  by  some  author- 
ities that  in  the  absence  of  such  currents  perfectly  pure  zinc  is  not 
attacked  by  dilute  acids. 


1  106.]  HEAT  OF  COMBINATION.  207 

The  density  of  the  acid  used  should  be  determined 
roughly  as  in  ^[  40. 

From  the  results  of  this  experiment  the  student 
is  to  calculate  (as  in  H  106,  below)  the  number  of 
units  of  heat  given  out  by  1  gram  of  zinc  in  uniting 
with  an  excess  of  dilute  nitric  acid,  also  what  part  of 
this  heat  is  due  to  its  uniting  with  the  oxygen  of  the 
acid.  The  heat  of  combination  of  zinc  with  nitric 
acid  will  be  found  to  have  an  important  bearing  upon 
problems  relating  to  electric  batteries  in  which  zinc 
is  the  dissolving  element  and  nitric  acid  the  oxidizing 
agent  (§  145). 

^|  106.  Calculations  relating  to  Heat  of  Combina- 
tion. —  It  is  necessary,  in  general,  to  find  the  spe- 
cific heat  of  the  liquid  used  for  a  determination  of 
heats  of  combination  (see  Experiment  34).  The 
specific  heats  of  certain  solutions,  amongst  them 
nitric  acid,  may  be  found,  when  their  densities  are 
known,  by  Table  30.  In  calculating  the  thermal 
capacity  of  a  calorimeter,  the  specific  heat  of  the 
glass  composing  the  inner  cup  may  be  taken  as 
0.19. 

If  wv  is  the  weight  of  zinc  employed,  ^  its  specific 
heat  (.09,")),  ^  its  original  temperature  •  if  w2  is  the 
weight  of  acid  employed,  s2  its  specific  heat  (from 
Table  30),  and  tz  its  original  temperature  reduced 
(see  ^[  93,  2)  to  the  time  of  solution  ;  if  c  is  the 
thermal  capacity  of  the  calorimeter,  ts  its  original 
temperature  (the  same  as  £2)  and  t  the  temperature  of 
the  mixture,  we  have  for  the  heat  of  combination  h 
(substituting  h  for  I  in  the  general  formula  of  ^[  100, 


208  CALORIMETRY.  [Exp.  38. 

and  changing  signs,  since  h  would  be  negative  if  heat 
were  absorbed), 

(w9  s2  +  e)  (£— O  +  w,  s,  (t—t.) 


If,  as  in  the  experiment,  a  comparatively  large  quan- 
tity of  acid  is  employed,  the  second  term  of  the  nu- 
merator may  be  neglected.  When,  moreover,  1  grain 
of  zinc  is  used,  w1  =  1,  and  we  have, 

h=(w2  s2  +  c)  (t  —  £2),  nearly.  II. 

The  truth  of  the  last  formula  is  sufficiently  evi- 
dent, since  s2  is  the  thermal  capacity  of  1  gram  of 
the  acid,  w2  s2  must  be  that  of  tv2  grams  ;  and  this 
added  to  the  thermal  capacity  (c)  of  the  calorimeter 
must  represent  (neglecting  the  1  gram  of  zinc)  the 
total  thermal  capacity.  In  the  formula  (II.)  the 
total  thermal  capacity  is  simply  multiplied  by 
the  number  of  degrees  rise  in  temperature.  This 
must  give  the  number  of  units  of  heat  developed  by 
the  combination  of  the  zinc  with  the  acid. 

The  heat  of  combination  of  zinc  oxide  may  be 
calculated  by  formula  I.  To  find  the  heat  given  out 
by  a  quantity  of  zinc  oxide  (1.25  grams,  nearly) 
which  contains  1  gram  of  metallic  zinc,  this  heat  of 
combination  must  be  multiplied  by  1.25.  The  same 
result  may  be  obtained  directly  by  formula  II.  if,  as 
in  the  experiment  described,  we  have  employed  1.25 
grams  of  zinc  oxide. 

The  chemical  reaction  which  takes  place  when 
zinc  is  dissolved  in  nitric  acid  may  be  divided  theo- 


T106.]  HEAT  OF  COMBINATION.  209 

retically  into  two  stages :  first,  the  combination  of  1 
gram  of  zinc  with  oxygen,  which  is  obtained  by  the 
decomposition  of  a  part  of  the  nitric  acid,1  thus  : 

Zinc.    Oxygen.      Zinc  oxide. 

Zn  +  O  =  Zn  O  ;  (1) 

and,  second,  the  combination  of  the  1.25  grams  of  zinc 
oxide  thus  formed  with  more  of  the  nitric  acid  to 
form  zinc  nitrate,  thus : 

Zinc  oxide.  Nitric  acid.  Zinc  nitrate.  Water 

ZnO  +  N2O5-H2O  =  ZnON2O5  +  H2O.  (2) 

We  have  already  found  the  heat  developed  by  the 
process  as  a  whole.  We  have  also  found  the  heat 
developed  in  the  second  stage  of  the  process,  namely, 
the  union  of  1.25  grams  of  zinc  oxide  with  nitric 
acid.  The  difference  between  these  two  quantities 
of  heat  must  (by  the  principle  of  the  conservation  of 
energy)  be  equal  to  the  heat  developed  by  1  gram  of 
zinc  in  combining  with  oxygen  extracted  from  nitric 
acid. 

If,  for  example,  1  gram  of  zinc  dissolving  in  100 
grams  of  nitric  acid  of  a  certain  strength  gives  out 

1  Nitric  acid,  thus  deprived  of  its  oxygen,  may  be  reduced  to  ni- 
trous acid,  nitric  oxide  (gas),  or  even  to  ammonic  nitrate.  The  reac- 
tions are  as  follows  :  — 

2  Zn  +  3  N205  •  H2O  =  2  ZnO  •  N2O5  +  2  H2O  +  N203  •  H2O  (nitrous 

acid). 

3  Zn  +  4  N205  •  H20  =  3  ZnO  •  N2O6  +  4  H2O  +  2  NO  (nitric  oxide). 

4  Zn  +  5  N.205  •  H20  =  4  ZnO  •  N205  +  3  H,O  +  ( H4N)  (NO3)  (ammo- 
nic nitrate). 

Nitrous  acid  may  be  formed  by  the  reduction  of  strong  nitric  acid. 
The  presence  of  nitric  oxide  gas  may  usually  be  recognized  by  the 
red  fumes  which  are  generated  when  nitric  acid  is  reduced.  Ammonic 
nitrate  is  formed  only  in  very  weak  solutions  (Wurtz,  Chimie 
Moderne,  p.  169). 


210 


CALORIMETRY. 


[Exp.  38. 


1,500  units  of  heat,  while  an  equivalent  (1.25  grams) 
of  zinc  oxide  gives  out  only  400  units  of  heat,  it  is 
evident  that  1500—400,  or  1100,  units  of  heat  are  due 
to  the  combination  of  1  gram  of  zinc  with  the  oxygen 
of  the  acid. 

Tf  107.  Heat  of  Combustion.  —  We  have  seen  in  the 
last  section  how  we  may  find  indirectly  the  amount 
of  heat  given  out  by  a  gram  of  a  given  material  when 
it  combines  with  the  oxygen  of  an  acid.  This  heat 
varies  greatly  according  to  the  difficulty  of  extracting 
the  oxygen  in  question.  If,  for  instance,  as  in  sul- 
phuric acid,  the  oxygen  must  be  taken  away  from 
hydrogen,  for  which  it  has  a  great  affinity,  nearly 
three  fourths  of  the  energy  will  be  spent  in  decom- 
posing the  acid.  In  the  case  of  nitric  acid,  less  diffi- 
culty is  encountered ;  since  nitric  acid  is  more  readily 
decomposed  (see  footnote,  ^[  106).  Even,  however, 
in  the  case  of  chromic  acid,  in  which  the  oxygen 
approaches  very  nearly  its  condition  in  the  free  state, 
the  heat  of  combination  with 
oxygen  will  differ  somewhat 
from  the  result  which  we 
should  obtain  by  burning  a 
metal  in  oxygen  gas. 

The  heat  given  out  by  one 
gram  of  a  substance  when 
burned  in  oxygen  is  called 
its  heat  of  combustion  in 
oxygen.  It  may  be  determined  directly  by  an  appa- 
ratus shown  in  Fig.  84.  The  substance  in  question 
is  placed  in  a  deflagrating  spoon,  f,  contained  in  a 


FIG.  84. 


•ff  107.]  HEAT  OF  COMBUSTION.  211 

water-tight  chamber,  h ;  oxygen  (or  air)  is  admitted 
to  this  chamber  by  the  tube  a,  and  the  gaseous  pro- 
ducts of  combustion,  if  any,  escape  through  the  spiral 
tube  gfc.  The  whole  system  of  tubes  is  surrounded 
by  water,  contained  in  a  calorimeter  of  the  ordinary 
sort.  When  the  temperature  of  the  water  has  been 
observed,  the  substance  is  ignited  by  a  current  of 
electricity.  From  the  rise  of  temperature  and  the 
thermal  capacities  of  the  calorimeter  and  its  contents, 
the  heat  of  combustion  is  calculated. 

To  determine  the  heat  of  combustion  of  a  gas  with 
this  apparatus,  a  third  tube  must  be  added  to  supply 
the  gas.  A  much  simpler  device  consists,  however, 
of  a  small  metallic  cone  soldered  into  the  bottom  of 
a  calorimeter.  The  cone  ends  above  in  a  spiral  tube, 
surrounded  by  water.  A  gas  jet  burned  beneath  this 
cone  will  give  up  nearly  all  of  its  heat  to  the  water. 
The  quantity  of  gas  used  is  measured  by  a  gas-meter. 
The  determination  of  heats  of  combustion  in  general 
is  an  exceedingly  difficult  problem,  but  the  ambitious 
student  may  be  encouraged  to  attempt  a  rough  deter- 
mination of  the  heat  of  combustion  of  coal-gas  or 
alcohol  with  a  simple  apparatus  like  the  one  de- 
scribed. 


212  RADIATION.  [Exp.  39. 


EXPERIMENT    XXXIX. 

RADIATION   OF    HEAT. 

^[  108.   The  Fyroheliometer.  —  A  simple  form  of  pyro- 
heliometer  (jrvp,  fire,  heat;  jjXios,  sun;  perpov,  meas- 
ure), or  instrument  for  measuring  the  heat  radiated 
by  the  sun,  consists  of  a  hollow 
tin  box  (Fig.  85)  filled  with  water. 
One  of  the  outer  surfaces  of  the 
box  is  blackened,  so  as  to  absorb 

most  of  the  heat  which  falls  upon 
1  IG.  85. 

it.  This  surface  is  turned  per- 
pendicularly to  the  rays,  the  intensity  of  which  is  to 
be  measured.  The  temperature  of  the  water  is  ob- 
served by  a  thermometer  passing  through  a  hole  in 
the  side  of  the  box.  The  number  of  heat  units  ab- 
sorbed is  calculated  from  the  rise  of  temperature  and 
thermal  capacity  of  the  vessel  and  its  contents,  as 
in  other  experiments  in  calorimetry.  An  allowance 
for  cooling  is  made  by  watching  the  thermometer 
when  the  instrument  is  in  shadow.  It  is  found  in 
this  way  that  the  solar  radiation  may  amount  to 
nearly  2  units  of  heat  per  minute  on  each  square 
centimetre  of  surface. 

The  p3rroheliometer  may  also  be  used  to  measure 
the  heat  radiated  by  a  candle,  or  any  other  source  of 
heat ;  or  it  may  be  employed  simply  to  compare  two 
sources  with  each  other.  In  all  such  experiments  it 
is  obvious  that  the  distance  of  a  given  source  of  heat 


«[  109.]  EFFECT   OF   DISTANCE.  213 

must  be  taken  into  account.  It  will  be  found,  for 
instance,  that  the  heat  radiated  by  an  ordinary  can- 
dle-flame at  a  distance  of  about  2  cm.  may  be  as 
intense  as  the  sun's  heat.  At  the  distance  of  a  deci- 
metre, the  heat  from  the  candle  could  hardly  be 
detected  by  a  pyroheliometer. 

^[  109.  Application  of  the  Law  of  Inverse  Squares. 
When  a  person  stands  midway  between  two  sources 
of  heat  which  are  equal  in  every  respect,  he  feels  of 
course  equal  intensities  of  radiation.  If,  however, 
one  of  these  sources  is  much  more  powerful  than  the 
other,  he  must  approach  the  smaller  of  the  two  in 
order  that  the  warmth  from  both  may  seem  to  be  the 
same.  Let  the  power  of  the  first  source  be  x,  and 
the  distance  from  it  a ;  let  the  power  of  the  second 
source  be  y,  and  the  distance  from  it  b  ;  then  accord- 
ing to  the  law  of  inverse  squares  (§  94)  the  effects  of 
the  two  sources  will  be  proportional  to  x  -f-  a2  and  to 
y  -5-  62,  respectively.  If  the  two  effects  are  equal,  it 
follows  that 

x  -T-  a2  =  y  -r-  b2 ;  or  x  :  y  : :  a2  :  b2. 

It  thus  appears  that  the  powers  of  any  two  sources  of 
radiant  heat  are  to  each  other  directly  as  the  squares 
of  the  distances  at  which  they  produce  equal  effects. 

The  same  reasoning  may  be  applied  to  two  sources 
of  light,  to  two  sources  of  sound,  or  to  any  two 
sources  of  radiant  energy,  the  effect  of  which  dimin- 
ishes as  the  square  of  the  distance  increases. 

We  have,  accordingly,  a  principle  by  which  we 
may  compare  any  two  sources  of  energy  of  the  same 


214  RADIATION.  [Exp.  39. 

kind  ;  namely  to  find  two  distances,  a  and  6,  at  which 
equal  effects  are  produced. 

To  test  the  equality  of  two  effects  with  any  degree 
of  precision,  it  is  necessary  to  employ  a  "differential" 
instrument  of  some  sort ;  that  is,  an  instrument  which 
is  constructed  especially  to  indicate  the  difference 
between  two  effects.  The  instrument  must  be  so 
delicate  that  in  the  absence  of  any  indication,  we 
may  assume  that  the  two  effects  are  equal.  The 
methods  for  the  comparison  of  two  sources  of  heat 
about  to  be  described,  will  be  found  to  belong  to  the 
general  class  known  as  "  null  methods  "  (§  42). 

^[  110.  The  Differential  Thermometer  and  the  Ther- 
mopile. —  I.  A  differential  thermometer,  useful  for  the 
comparison  of  two  sources  of  radiant  heat,  may  be 
constructed  as  follows:  two  cylindrical  metallic 

boxes,  d  and  e,  about 
10  cm.  in  diameter,  and 
1  cm.  deep,  are  made 
out  of  the  thinnest 
brass,  and  fastened  by 
a  layer  of  wax  to  the 
support  bh.  The  glass 
FlG>  86>  U-tube  or  gauge,  fg, 

contains  a  little  colored  liquid,  and  is  attached  by 
rubber  couplings  to  the  boxes  d  and  e,  so  that  the 
system  may  be  air-tight.  The  outer  faces  of  the  boxes, 
d  and  e,  are  coated  with  lampblack,  to  absorb  heat ; 
the  sides  may  be  covered  with  wool  to  prevent  loss 
of  heat.  The  two  conical  shields,  a  and  <?,  blackened 
inside,  are  finally  added  to  cut  off  lateral  radiation. 


I  no.] 


THE  THERMOPILE. 


215 


A  very  slight  amount  of  heat  falling  on  the  black- 
ened surface  of  either  of  the  cylinders,  d  or  e,  will 
cause  an  expansion  of  air  within  the  cylinder  in  ques- 
tion. Unless  this  is  offset  by  an  equal  expansion  of 
air  due  to  an  equal  amount  of  heat  falling  on  the 
other  cylinder,  the  level  of  the  liquid  in  the  gauge 
fg  will  be  affected. 

II.  An  instrument  which  may  be  made  much  more 
sensitive  than  a  differential  thermometer  is  repre- 
sented in  Fig.  87,  and  in  de,  Fig.  88. 
It  consists  of  an  alternate  series  of  strips 
of  bismuth  and  antimony,  joined  to- 
gether in  a  sort  of  zigzag.  Only  four 
strips  are  shown  in  the  figure,  but  a 
much  greater  number  is  generally  used.  The  com- 
bination is  known  as  a  "  thermopile,"  or 


FlG-  87- 


heat-bat- 


tery."   It  is  usually  mounted  on  a  support  (Fig.  88), 


FIG. 


and  provided  with  two  conical  shields,  a  and  c. 
When  heat  falls  on  either  set  of  junctions,  as  d, 
a  current  of  electricity  is  generated  (see  Exp.  95). 
This  current  is  measured  by  a  galvanometer,/,  the 


£16  EADIATION.  [Exp.  39. 

terminals  of  which  are  connected  by  wires  with  the 
terminals  of  the  thermopile.  The  deflection  of  the 
galvanometer  needle  is  reversed  if  heat  falls  on 
the  opposite  face  of  the  thermopile,  e.  When  equal 
amounts  of  heat  fall  on  both  the  faces,  d  and  e,  the 
needle  should  not  be  deflected. 

It  would  be  out  of  place  here  to  discuss  the  prin- 
ciples which  underlie  the  phenomena  in  question. 
The  student  should  for  the  present  regard  a  thermo- 
pile and  galvanometer  simply  as  a  convenient  substi- 
tute for  a  differential  thermometer  and  U-tube. 


FIG.  89. 

^[  111.  Determination  of  Candle-Heat-Power. — A 
thermopile  connected  with  a  galvanometer,  as  in  Fig. 
88,  is  mounted  on  a  fixed  support  (be,  Fig.  89),  in 
the  middle  of  a  horizontal  graduated  rail  (gli).  The 
needle  of  the  galvanometer  is  made  to  point  to  zero 
(^[  112,  7).  Two  movable  supports,  d  and  /,  con- 
structed so  as  to  slide  along  the  rail,  are  placed  one 
on  each  side  of  the  thermopile.  A  candle  (a)  and 
a  small  kerosene  lamp  (<?)  are  then  mounted  on  the 


IT  111.]  THE   THERMOPILE.  217 

supports,  d  and  f  respectively,  so  that  the  flames 
may  be  on  a  level  with  the  thermopile  (^f  112,  5). 
The  supports  are  then  to  be  set  permanently  at  such 
distances  from  the  thermopile  (^[  112,  2)  that  either 
flame  alone  will  cause  a  deflection  of  the  galvanom- 
eter of  at  least  45°  (^[  112,  1),  but  that  both  together 
will  cause  little  or  no  deflection.  The  height  of  the 
lamp-flame  is  then  adjusted,  if  necessary,  until  the 
deflection  is  reduced  to  zero. 

The  lamp  and  candle  while  still  burning  are  next 
to  be  weighed  as  accurately  as  possible  on  a  pair  of 
open  scales  (Fig.  1,  ^[  2),  and  the  time  of  weighing 
is  to  be  noted  in  each  case.  The  lamp  and  candle 
are  then  returned  to  their  former  positions  on  the 
supports  d  and  /,  where  they  are  allowed  to  burn  for, 
let  us  say,  half  an  hour. 

Meanwhile  the  distance  of  each  from  the  nearer 
face  of  the  thermopile  is  accurately  determined  by 
means  of  the  markers  (#  and  A),  which  should  be 
just  under  the  centres  of  the  flames  (^[  112,  3).  The 
distance  (de,  Fig.  88)  between  the  faces  of  the  ther- 
mopile must  also  be  measured  and  allowed  for  (^[  112, 
4).  If  the  needle  of  the  galvanometer  shows  any 
deflection  in  the  course  of  the  experiment,  it  must  be 
brought  back  to  zero  by  increasing  or  diminishing 
the  flame  of  the  lamp.  At  the  end  of  the  half-hour, 
the  candle  and  lamp  are  to  be  re-weighed  in  the  same 
order  as  before,  while  still  burning. 

The  candle  and  lamp  are  now  to  be  replaced  on 
their  supports  (<2  and  /  respectively),  each  of  which 
is  to  be  set  permanently  at  the  same  distance  from 


218  RADIATION.  [Exp.  39. 

the  thermopile  as  before,  but  on  the  other  side  of  it 
(1[  112,  8).  The  height  of  the  lamp-flame  is  to  be 
adjusted  so  as  to  neutralize  the  heat  from  the  candle  ; 
and  at  the  end  of  another  half-hour,  the  lamp  and 
candle  are  to  be  re-weighed,  as  before,  while  still 
burning. 

Instead  of  a  thermopile,  a  differential  thermometer 
(T[  110)  may  be  employed,  with  essentially  the  same 
precautions  (see  ^[  112).  Instead  of 
the  kerosene  lamp,  an  electric  incandes- 
cent lamp  may  be  used  (Fig.  90).  In 
this  case  it  is  necessary  that  the  zinc- 
plates  of  the  battery  furnishing  the 
electricity  for  the  lamp  should  be 
weighed  before  and  after  the  experi- 
ment. These  plates  should  be  well 
amalgamated  with  mercury  to  prevent 
unnecessary  loss  of  material. 

In  any  case  the  candle-heat-power  of 
the  lamp  is  to  be  calculated  and  reduced  to  the 
standard  rate  of  consumption,  as  will  be  explained 
in  T[  113. 

^[  112.  Precautions  in  the  Determination  of  Candle- 
Power. —  (1)  Before  attempting  an  accurate  com- 
parison of  two  sources  either  of  heat  or  of  light,  it  is 
well  to  make  sure  that  the  instrument  to  be  employed 
is  sufficiently  sensitive  (§  42).  For  this  purpose  it  is 
first  exposed  to  the  radiation  from  the  feebler  source 
alone.  To  make  a  comparison,  for  instance,  accurate 
within  1  per  cent,  the  response  must  be  100  times 
as  great  as  the  minimum  perceptible.  The  sensi- 


T  112.]  PRECAUTIONS.  219 

tiveness  of  the  combination  should,  if  necessary, 
be  increased  by  bringing  the  source  in  question 
closer  to  the  instrument  until  a  sufficient  response 
is  obtained. 

(2)  It  is  important  that  one  of  the  two  sources 
compared  should   be  at   a  fixed  distance   from  the 
instrument   throughout  an   experiment.     When  an 
oil  lamp  or  gas-flame  is  one  of  the  sources,  so  that 
the  height  of  the  flame  can  be  adjusted,  it  is  well 
that   both   sources   should   be   fixed ;    and   for   con- 
venience in  calculation,  each  distance  may  be  made 
equal  to  some  round  number. 

(3)  The  distance  of  the  sources  from  the  instru- 
ment may  be  most  conveniently  determined  by  means 
of  markers  (#,  A,  in  Fig.  89).     These  markers  should 
be  in  line  with  the  centre  of  the  source  of  light  or 
heat  (as,  for  instance,  A),  not  at  one  side  of  it  (like  g). 
The  student  should  confirm  the  indications  of  the 
markers  by  direct  measurements.     It  should  be  re- 
membered that  the  distances  sought  lie  between  the 
centre  of  a  flame  and  the  surface  illuminated  by  it. 

(4)  Care  must  be  taken  in  measuring  the  distances 
ad  and  ec  to  allow  for  the  distance  de  (Figs.  86  and 
88)  between  the  two  surfaces  illuminated.     This  dis- 
tance should  be  determined  by  direct  measurement ; 
for  this  purpose  the  conical  shields  must  of  course  be 
removed. 

(5)  It  is  important  that  the  rays  of  light  or  of 
heat  should  be  equally  inclined  with  respect  to  the 
two  surfaces  d  and  e.     To  help  in  securing  this  re- 
sult, the  surfaces  should  be  made  vertical,  and  the 


220  RADIATION.  [Exp.  39. 

sources  of  light  or  heat  should  be  raised  or  lowered 
until  they  are  on  a  level  with  these  surfaces.  Neither 
angle  of  incidence  should  exceed  20°.  In  this  case 
slight  differences  in  the  angles  of  incidence,  as  in 
Figs.  96  and  98,  will  have  no  perceptible  effect  on 
the  result. 

(6)  The  conical  shields  a  and  b  (Figs.  86  and  88) 
will   serve  to  cut  off  lateral  radiation.     It  is,  how- 
ever, necessary  to  place  large  black  screens  behind 
two  sources  of  light  which  are  being  compared,  so  as 
to  shut  out  light  from  all  other  sources.      A  dark 
room  is  of  great  service  in  photometry ;   a  room  of 
uniform  temperature  is  equally  important  in  measure- 
ments of  radiant  heat. 

(7)  Before   comparing    two   sources    of    heat  or 
light,  it  is  well  to  make  sure  that  the  instrument 
to  be  employed  is  not  affected  by  radiation  from  the 
windows  or  from  the  walls  of  the  room  (§  32).     The 
liquid  in  a  differential  thermometer  should  stand  at 
the  same  level,   for  instance,  in  both  arms   of  the 
gauge.     If  it  does  not,  the  gauge  should  be  tempo- 
rarily disconnected  so  that  the  air-pressure  may  be 
equalized.     The  needle  of  a  galvanometer  connected 
with  a  thermopile  should  point  to  zero,  otherwise  it 
should  be  made  to  do  so  by  twisting  the  thread  by 
which  it  is  suspended,  or  by  placing  a  magnet  in  its 
neighborhood.     If  the  two  surfaces  of  a  photometer 
do  not  appear  equally  dark,  it  is  necessary  to  make 
a  rearrangement   of  the  screens,  by  which  at  least 
equality  of  illumination  may  be  secured. 

(8)  To  eliminate  all  errors  arising  from  unequal 


1  113.]  CANDLE-POWER.  221 

radiation  from  surrounding  objects,  and  from  any 
inequality  in  the  surfaces  illuminated,  two  determi- 
nations should  always  be  made  (see  §  44).  In  one  of 
these  a  given  surface  is  illuminated  by  the  weaker 
source  of  light  or  heat ;  in  the  other,  it  is  illuminated 
by  the  stronger  source.  An  error  in  the  adjustment 
of  the  markers  may  also  be  eliminated  in  this  way. 

^[  113.  Calculations  relating  to  Candle-Power.  — 
The  standard  candle  is  denned  as  one,  seven-eighths 
of  an  inch  in  diameter  (six  to  the  pound),  burning 
120  grains  of  spermaceti  per  hour.  A  paraffine  can- 
dle does  not  give  out  quite  so  much  light  as  a  sperm 
candle  under  similar  circumstances.  It  is  thought 
that  no  perceptible  error  will  be  committed  by  substi- 
tuting for  a  standard  candle  one  of  paraffine  burning 
8  grams  per  hour  (123|-  grains,  nearly).  An  ordinary 
candle  may  of  course  burn  a  little  more  or  less  than 
the  standard.  Since  the  heat  or  the  light  is  very 
nearly  proportional  to  the  rate  of  consumption,  we 
find  that  the  actual  candle-power  of  a  paralfine  can- 
dle 1  is  equal  to  one  eighth  the  weight  in  grams  of 
the  paraffine  burned  in  one  hour.  This  gives  us  the 
quantity,  x,  in  the  formula  of  ^[  109.  Hence,  if  a 
lamp  at  a  distance  b  has  the  same  effect  as  x  standard 
candles  at  the  distance  a,  as  regards  either  heat  or 
light,  we  may  find  the  number  of  standard  candles,  y, 
to  which  this  lamp  is  equivalent  by  the  formula  — 

J2 

y  =  -o  x. 
9      a2 

1  The  heat  radiated  in  all  directions  by  an  ordinary  candle  amounts 
to  about  2  units  per  second.  This  is  only  a  small  part  of  the  total 


222  KADIATION.  [Exp.  40. 

By  the  "  candle-power "  of  a  lamp  is  ordinarily 
meant  the  number  of  standard  candles  to  which  it  is 
equivalent  in  respect  to  light  (see  Exp.  40).  The 
number  of  candles  to  which  it  is  equivalent  in  respect 
to  the  radiation  of  heat  may  be  called  its  "  candle- 
heat-power."  It  is  evident  that  the  thermopile  and 
the  differential  thermometer,  which  absorb  all  rays 
alike  (whether  visible  or  invisible),  are  instruments 
for  determining  the  candle-heat-power  as  distinguished 
from  the  candle-light-power  of  any  source. 

It  is  interesting  to  reduce  the  candle-power  of  a 
lamp  to  the  normal  rate  of  consumption  of  a  candle 
(8  grams  per  hour).  We  first  divide  the  actual  can- 
dle-power of  the  lamp  by  the  number  of  grains 
burned  in  one  hour  to  find  the  candle-power  corres- 
ponding to  1  gram  per  hour ;  then  we  multiply  the 
result  by  8.  A  surprising  similarity  exists  between 
the  candle-powers  of  different  materials  when  thus 
reduced  to  a  common  standard. 


EXPERIMENT  XL. 

PHOTOMETRY. 

^[  114.    Determination   of  Candle-Power  by  means  of 
a    Photometer.  —  I.    BUNSEN'S    PHOTOMETER.  —  A   Very 

fair  comparison  of  two  sources  of  light  may  be  made 
by  means  of  a  scrap  of  white  paper  rendered  trans- 
quantity  of  heat  generated  by  combustion,  which  amounts  to  about 
20  units  per  second.  Less  than  4  %  of  the  radiant  heat  is  visible  as 
light 


BUNSEN'S  PHOTOMETER. 


223 


lucent  at  the  centre  by  a  drop  of  oil  or  varnish. 
When  such  a  scrap  is  held  up  in  front  of  a  light,  the 
oil-spot  appears  bright,  as  in  Fig.  91 ;  when  held 
behind  a  light,  it  looks  dark,  as  in  Fig.  92.  If  both 


FIG.  91. 


FIG.  92. 


sides  of  the  paper  are  equally  illuminated,  the  spot 
may  nearly  or  quite  disappear.  Usually,  however, 
the  oil-spot  seems  a  little  darker  than  the  rest  of  the 
paper.  It  is  necessary,  therefore,  to  look  at  it  from 
both  sides.  When  it  appears  equally  dark  from 
both  points  of  view,  we  may  infer  that  the  two  sides 
of  the  paper  are  equally  illuminated. 

To  make  use  of  an  oil-spot  for  a  comparison  of 
two  lights,  the  paper  (5,  Fig.  93)  is  provided  with 


two  shields,  a  and  <?,  to  cut  off  lateral  radiation,  and 
is  mounted  in  the  place  of  the  thermopile  (6,  Fig.  89, 
Tf  111)  between  a  candle,  a,  and  a  lamp,  c.  The 
lamp-flame  is  adjusted  as  in  ^[  111  until  the  paper 
seems  equally  illuminated  on  both  sides,  d  and  e. 


224  RADIATION.  [Exp.  40. 

The  distances  of  the  lamp  and  candle,  and  the  weights 
burned  in  one  hour  by  each  are  found  in  the  same 
manner  as  with  the  thermopile. 

In  practice  the  form  given  to  the  shields  is  not 
generally  conical,  as  in  the  case  of  a  thermopile,  but 
barrel-shaped  (see  Fig.  94).  The  object  of  this  is  to 
facilitate  the  examination 
of  the  oil-spot  through 
two  openings,  a  and  c. 
Such  an  instrument  is 
called  a  Bunsen's  pho- 


FIG.  94. 

The  general  precau- 
tions in  the  use  of  a  photometer  have  already  been 
enumerated  (^[  112).  Certain  special  precautions 
will  be  considered  in  ^[  115.  The  results  are  to  be 
reduced  as  in  ^[  113. 

II.  RUMFORD'S  PHOTOMETER.  —  If  the  diaphragm 
and  shields  used  in  Bunsen's  photometer  (Fig.  93) 
are  removed,  leaving  only  the  rod  by  which  they 
were  supported,  and  if  a  piece  of  paper  (ae,  Fig.  95) 
is  fastened  to  this  rod  so  as  to  be  equally  inclined  to 
the  rays  falling  upon  it  from  the  lamp  and  from  the 
candle  (Fig.  89) ;  then  when  the  flames  are  placed 
at  such  distances  as  to  give  equal  amounts  of  light  at 
the  point  6,  the  shadows  ab  and  be 
(Fig.  95)  cast  by  the  rod  should 
be  equally  dark.  The  instrument, 
thus  arranged,  is  a  form  of  Rnrriford's 
photometer,  depending  upon  the 
principle  that  equal  illuminations  cause  equal  shad- 


o 

Ij,.       I    —  —  :• 


IT  114,  II.]  RUMFORD'S  PHOTOMETER.  225 

ows ;  it  might  be  substituted  for  a  Bunsen's   pho- 
tometer for  a  rough  comparison  of  two  lights. 

It  is  obvious,  however,  that  a  slight  inclination  of 
the  paper  might  expose  it  very  unequally  to  the  rays 
from  the  two  sources,  and  thus  vitiate  the  results. 
To  lessen  errors  from  this  source,  both  lights  are  in 
practice  placed  on  the  same  side  of  the  rod,  b  (Fig. 
96),  the  two  shadows  of  which,  d  and  e^  are  thrown 
horizontally  on  the  vertical  surface,  de.  When  these 
shadows  have  been  made  equally  dark  by  adjust- 
ing the  distances  of  the  lamp  and  candle,  or  the 
height  of  the  lamp-flame  the  two  lights  are  to  each 


FIG.  96. 


other  as   the  squares  of  the  distances   ae   and   cd. 
These  distances  are  therefore  to  be  measured. 

The  student  should  observe  that  the  distance  of  the 
rod  from  the  screen  may  affect  the  sharpness  of  the 
shadows,  but  not  their  darkness,  which  depends  simply 
on  the  distance  of  the  lights  from  the  screen.  It  is 
well  to  have  the  rod  close  to  the  screen,  in  order  that 
the  two  shadows  may  be  near  together,  but  not  so 
close  that  the  shadows  overlap.  A  small  amount  of 
light  from  the  windows  need  not  vitiate  the  result, 
provided  that  it  casts  no  shadow  on  the  screen.  If 
it  does,  the  light  must  be  cut  off. 

15 


226 


RADIATION. 


[Exp.  40. 


The  weights  burned  by  the  lamp  and  candle  in 
one  hour  are  found  as  with  a  Bunsen's  photometer 
(I.),  or  with  a  thermopile  (^[  111)  ;  and  the  results 
are  reduced  in  the  same  manner  (  ^[  113). 

III.  Box  PHOTOMETER.  —  Instead  of  using  a  rod, 
as  in  Rumford's  photometer  (Fig.  96),  it  is  sometimes 

advantageous  to  em- 
ploy a  partition  (5,  Fig. 
97).  One  half  (d)  of 
the  screen,  c?e,  may 
thus  be  illuminated  by 
the  candle  (a),  and  the 
other  half  (e)  by  the 
lamp  (&).  The  screen 
is  made  translucent,  so  that  the  intensities  of  illu- 
mination may  be  compared  with  the  eye  behind  it. 

This  form  of  photometer  is  particularly  useful 
when  it  is  possible  to  enclose  the  whole  apparatus  in 
a  box.  A  horizontal  section  of  such  a  box  is  shown 


FIG  97. 


t 


FIG.  98. 

in  Fig.  98.     The  distances  ad  and  ce  are  measured 
directly  by  a  metre  rod. 

If  the  angles  of  incidence,  adb  and  ceb,  differ  by 
more  than  10°,  it  may  be  well  to  alter  the  screen  de 
slightly  so  that  its  inclination  to  both  rays  may  be 
the  same  flf  112,  5). 


1  114,111.] 


BOX  PHOTOMETER. 


227 


An  arrangement  in  which  the  distances  of  the  lamp 
and  candle  are  adjustable  is  represented  in  Fig.  99, 
which  gives  a  view  of  the 
apparatus  from  above. 
The  lights  are  contained 
each  in  one  of  the  sliding 
boxes,  e  and  /.  The  top 
of  the  main  box  is  closed 
as  far  as  the  ends  of  the 
sliding  boxes  by  a  set  of 
covers  (6,  c,  and  d~).  All 
direct  light  is  thus  ex- 
cluded from  the  photome- 
ter. A  cloth  cover,  a,  may 
be  thrown  over  the  head 
when  it  is  desired  to  com- 
pare very  feeble  illumina- 
tions. 

Box  photometers  may 
also  be  constructed  on 
Bunsen's  or  on  Rumford's 
principle.  They  have  the 
advantage  of  a  dark  room 
without  its  expense  or  in- 
convenience. 

The  determinations  of 
candle-power,  and  the  re- 
duction of  the  results,  are 
made  in  precisely  the  same 
manner  as  in  II.  with  a  Rumford's  photometer 
also  1[  113. 


FIG.  99. 


See 


228  RADIATION.  [Exr-.  40. 

^[115.  Errors  in  Photometry  due  to  Color  Blind- 
ness.—  Light  is  essentially  a  physiological  as  distin- 
guished from  a  physical  quantity.  There  is  no 
standard  by  which  we  may  prove  that  one  kind  of 
light  is  more  brilliant  than  another.  A  person  who  is 
"color-blind"  may  consider  a  blue  light  brighter  than 
a  red  light,  which  to  a  person  of  "  normal  vision  " 
may  seem  much  the  brighter  of  the  two.  All  eyes 
are  in  a  certain  sense  color-blind,  since  the  greater 
part  of  the  rays  which  fall  upon  them  are  wholly 
invisible. 

The  modern  theory  of  color  may  be  stated  briefly 
as  follows:  There  are  three  principal  effects  pro- 
duced on  the  eye  by  rays  of  light.  The  first  is  to 
excite  in  the  retina  a  sensation  which  we  call  red. 
This  is  due  mostly  to  waves  of  light  between  60  and 
70  millionths  of  a  centimetre  in  length.  The  second 
is  to  excite  a  sensation  which  we  call  green.  Nearly 
all  rays  of  light  produce  this  effect  (green)  to  a  cer- 
tain extent ;  but  it  is  caused  most  strongly  by  waves 
between  50  and  60  millionths  long.  The  third  effect 
is  a  sensation  which  we  call  violet,  due  to  waves 
from  40  to  50  millionths  in  length.  When  waves1 
of  different  lengths  are  mixed,  complex  sensations 
are  produced.  Red  and  green  rays  together  may 
produce,  for  instance,  a  sensation  which  we  call  yel- 
low ;  violet  and  green  may  produce  blue  ;  red  and 
violet  may  produce  purple  ;  while  red,  green,  and 

1  The  student  must  distinguish  carefully  the  effects  of  mixing 
waves  of  light  from  the  effects  of  mixing  paints.  These  effects  are 
in  a  certain  sense  opposite. 


I  115.]  COLOR  BLINDNESS.  229 

violet  rays  together  may  cause  the  sensation  which 
we  are  familiar  with  in  ordinary  white  light.  Again, 
a  single  wave  may  produce  two  sensations :  one  60 
millionths  of  a  centimetre  long  will,  for  instance, 
produce  the  double  sensation  which  we  call  yellow  ; 
while  one  50  millionths  long  will  appear  blue.  The 
various  hues  which  we  find  in  different  objects  are 
due  to  the  proportions,  simply,  in  which  the  sensa- 
tions of  red,  green,  and  violet  are  excited.  The  eye 
is  capable  of  no  fourth  sensation  by  which  the  effect 
can  be  modified.  According  to  this  theory,  two  lights 
should  be  compared,  (1)  by  means  of  the  red  rays, 
(2)  by  means  of  the  green  rays,  and  (3)  by  means 
of  the  violet  rays  which  they  emit. 

The  simplest  way  to  compare  the  candle-power  of 
two  lights  with  respect  to  red  rays  is  to  hold  a  piece 
of  ordinary  "  ruby  glass  "  before  the  eye  in  observing 
the  brilliancy  of  the  two  surfaces  illuminated.  Green 
and  violet  glasses  may  similarly  be  employed  for  the 
green  and  violet  rays ;  but  pure  violet  glass  can 
hardly  be  obtained.  It  is  better  to  use  a  piece  of 
ordinary  glass  stained  with  violet-aniline  containing 
a  trace  of  Prussian  blue. 

With  these  precautions,  personal  errors  in  photo- 
metry might  undoubtedly  be  diminished,  particularly 
in  the  comparison  of  lights  of  different  hues  or  tints  ; 
but  as  long  as  the  eye  alone  is  used  to  compare  the 
brilliancy  of  two  surfaces,  it  is  doubtful  whether  the 
errors  of  a  photometric  comparison  can  ever  be 
greatly  reduced.  The  "  probable  error  "  of  such  a 
.comparison  may  be  estimated  at  about  5  per  cent. 


230  FOCAL  LENGTHS.  [Exr.  41. 

EXPERIMENT  XLL 

PRINCIPAL    FOCI. 

^f  116.  Determination  of  the  Principal  Focal  Length 
of  a  Converging  Lens.  —  The  principal  focal  length  of 
a  lens  may  be  defined  (see  §  103)  as  the  distance  at 
which  it  brings  parallel  rays  to  a  focus.  An  "  optical 
bench,"  convenient  for  tjie  measurement  of  focal 
lengths,  is  represented  in  Fig.  100.  It  consists  of 
a  wooden  plank,  set  up  edgewise,  with  three  sliding 
supports,  d*  e,  and/,  the  positions  of  which  are  deter- 


mined respectively  by  the  markers  #,  A,  and  t.  The 
apparatus  is  in  fact  the  same  as,  or  similar  to,  one 
already  employed  in  Experiments  39  and  40  (see 
Fig.  89). 

(1)  ORDINARY  METHOD.  —  To  find  the  principal 
focal  length  of  a  lens,  it  is  mounted  (see  b,  Fig.  100) 
on  one  of  the  slides  (e),  directly  over  the  marker 
(A)  (see  ^[  112,  3)  ;  and  a  translucent  screen  (c?)  is 
attached  to  another  slide  (/)  directly  over  the 
marker  (f).  The  third  slide  (<f)  is  temporarily  re- 
moved, so  that  the  rays  from  distant  points  (at  the 


IF  116  (1).]  PRINCIPAL  FOCI.  231 

left  of  the  figure)  may  be  focussed  by  the  lens  (5)  on 
the  screen  (c).  That  this  may  be  possible,  the  bench 
should  be  set  up  in  front  of  an  open  window  com- 
manding a  distant  view.1  Either  houses  or  trees 
may  afford  suitable  images.  It  is  assumed,  however, 
that  the  objects  in  question  are  so  far  off  that  rays 
from  any  point  in  these  objects  may  be  considered 
parallel.  They  should  be  at  least  a  hundred  times  as 
far  from  the  lens  as  the  lens  is  from  the  screen. 

The  distance  between  the  lens  and  the  screen  is  to 
be  adjusted  so  that  the  image  thrown  on  the  screen 
may  be  as  distinct  as  possible.  The  image  may  be 
viewed  either  from  in  front  or  (since  the  screen  is 
translucent)  from  behind.  The  number  of  details 
visible  in  the  image  is  the  test  of  its  distinctness  most 
easily  applied.  When  difficulty  is  found  in  the  pre- 
cise adjustment  of  distance,  the  screen  is  first  brought 
so  near  the  lens  that  the  most  minute  details  dis- 
appear; then  it  is  placed  so  far  from  the  lens  that  the 
same  result  is  obtained.  Midway  between  these  two 
positions  is  the  principal  focus  of  the  lens. 

The  distance  of  the  principal  focus  from  the  centre 
of  the  lens  is  taken  as  tho  measure  of  its  principal 
focal  length.  It  is  determined  by  observing  the  posi- 
tions of  the  two  markers,  h  and  z,  with  respect  to  the 
scale  close  behind  them.  If  either  of  the  markers  is 
out  of  line  with  the  lens  or  screen,  as  the  case  may  be, 
an  error  will  evidently  be  introduced  into  the  result 

1  In  the  absence  of  any  suitable  object,  we  may  use  a  projecting 
lantern,  focussed  so  as  to  give  parallel  rays.  To  obtain  this  result, 
the  slide  must  be  placed  in  the  principal  focus  of  the  projecting  lens. 


232  FOCAL  LENGTHS.  [Exp.  41. 

(^[  112,  3).  To  eliminate  this  error,  we  may  inter- 
change the  places  of  the  lens  and  screen.  The  whole 
bench  must  then  be  turned  round  so  that  an  image 
may  be  formed  by  the  lens  on  the  back  of  the  screen. 
The  thickness  of  the  screen  should  be  so  small  that 
it  need  not  be  taken  into  account.  If  either  of  the 
markers  is  out  of  line,  the  distance  between  the  lens 
and  screen  will  apparently  be  increased  in  one  case 
but  diminished  in  the  other  case,  and  by  an  equal 
amount.  The  average  of  the  two  distances  indicated 
by  the  markers  is,  therefore,  the  true  distance  from 
the  centre  of  the  lens  to  the  screen. 

If  there  is  a  second  scale  on  the  farther  side  of  the 
bench,  there  will  be  no  need  of  turning  it  round. 
We  have  only  to  turn  round  the  slides  e  and/. 

It  is  well  to  confirm  the  accuracy  of  the  scale  or 
scales  in  question  by  a  direct  measurement  between 
the  thin  edge  of  the  lens  and  the  screen.  The  mea- 
suring rod  must  be  held  perpendicular  to  the  screen, 
as  in  Fig.  100.  One  measurement  should  be  taken 
from  the  farther  edge  of  the  lens,  another  from  the 
nearer  edge,  and  a  third  from  the  top  of  the  lens.  If 
any  marked  differences  are  observed,  the  lens  should 
be  readjusted  until  these  differences  disappear. 

(2)  METHOD  OF  PARALLAX.  —  Instead  of  using  a 
screen  (c,  Fig.  100),  we  may  employ  a  wire  netting 
or  simply  a  vertical  wire.  If  the  wire  coincides  in 
position  with  the  image  formed  by  the  lens,  no 
"  parallax  "  (§  25)  will  be  apparent  when  the  eye  is 
moved  from  side  to  side.  If  the  wire  is  behind  the 
image,  it  will  seem  to  follow  the  eye ;  or  if  it  is  in 


1116(3).]  .          PRINCIPAL  FOCI.  233 

front  of  it,  it  will  always  appear  to  move  in  the  oppo- 
site direction  (see  diagrams,  Fig.  103,  ^[  118).  The 
phenomena  of  parallax  afford  in  fact  a  very  delicate 
test  by  which  a  wire  may  be  placed  exact!}7  in  the 
image,  and  the  position  of  the  image  thus  accurately 
determined.  This  is  called  focussing  by  the  method 
of  parallax.  The  distance  of  the  image  from  the  lens 
is  found  from  the  indications  of  the  markers,  and 
confirmed  by  direct  measurements  as  before  (see  1). 

(3)  INDIRECT  METHOD.  —  Another  way  of  finding 
the  principal  focus  of  a  lens  involves  the  use  of  a  tele- 
scope, which  has  _ 
been  adjusted  so 
that  parallel  rays 
striking  the  object-  FlG' 101' 
glass  (g,  Fig.  101)  are  brought  to  a  focus  at  a  point  o 
where  cross-hairs  are  placed.  The  first  step  in  focus- 
sing a. telescope  is  always  to  make  the  distance  of 
the  eye-piece  (b)  from  the  cross-hairs  (c)  such  that  the 
latter  may  be  seen  as  clearly  as  possible  through  the 
opening  a.  This  is  done  by  sliding  the  tube  d  within 
the  tube  e.  Then  the  tube  e  is  pushed  into  or  drawn 
out  from  the  tube /so  that  the  cross-hairs  may  coin- 
cide with  the  image  at  c.  In  the  last  adjustment, 
care  must  be  taken  not  to  disturb  the  distance  of  the 
eye-piece  from  the  cross-hairs,  unless,  as  sometimes 
happens,  the  focus  of  the  eye  has  changed  so  that 
the  cross-hairs  are  no  longer  visible  ;  in  this  case  the 
first  adjustment  must  be  repeated  before  the  second 
can  be  made.  In  some  telescopes  the  method  of 
focussing  by  parallax  (see  2)  can  be  used,  but  gen- 


234  FOCAL  LENGTHS.  .  [Exp.  41. 

erally  we  have  to  depend  simply  on  the  distinctness 
of  the  image  (see  1).  If  the  telescope  is  accurately 
focussed,  the  image  and  the  cross-hairs  should  both 
appear  distinct  to  the  eye. 

A  telescope  thus  focussed  is  mounted  as  in  Fig. 
100  at  any  point,  a,  in  front  of  a  lens,  b.  It  will  prob- 
ably be  found  that  a  page  of  fine  print  replacing  the 
screen,  c,  may  be  easily  read  through  the  combination. 
The  distance  of  the  page  from  the  lens  should  be 
varied  if  necessary,  so  that  the  print  may  seem  as 
distinct  as  possible. 

The  student  should  note  that,  owing  to  the  parallel- 
ism of  the  rays  from  a  given  point  in  passing  between 
the  lens  and  the  telescope,  the  distance  between  the 
lens  and  telescope  does  not  affect  the  focus. 

The  principal  focal  length  of  a  lens  has  been  defined 
as  the  distance  from  the  lens  at  which  parallel  rays 
are  brought  to  a  focus ;  it  might  also  have  been  de- 
fined as  the  distance  from  an  object  at  which  rays 
diverging  from  it  are  rendered  parallel  by  the  lens. 
It  is  evident  that  the  rays  diverging  from  any  point 
of  the  printed  page  (c)  must  be  rendered  parallel  by 
the  lens  (6)  in  order  to  be  visible  in  the  telescope 
(a)  ;  for  this  telescope  has  been  focussed  for  parallel 
rays,  and  cannot,  therefore,  be  in  focus  for  any  others. 
It  follows  that  the  distance  from  the  lens  to  the 
screen  is  equal  to  the  principal  focal  length  of  the 
lens;  the  latter  is,  therefore,  to  be  measured  as  in 
the  methods  previously  described  (see  1  and  2). 

(4)  COLOR  METHOD.  —  Instead  of  depending  en- 
tirely upon  the  distinctness  with  which  the  print  can 


T  116  (4).]  PRINCIPAL  FOCI.  235 

be  read,  we  may  observe  the  colors  with  which  each 
black  letter  seems  to  be  surrounded.  Unless  the  lens 
is  of  peculiar  construction,  so  as  to  focus  all  rays 
alike,  it  will  be  found  impossible  to  avoid  this  phe- 
nomenon. Let  us  suppose  that  the  red  rays  are 
accurately  focussed  ;  then  the  green  and  violet  rays 
will  be  just  out  of  focus,  and  hence  somewhat  scat- 
tered. The  spaces  which  would  otherwise  be  per- 
fectly black  will,  therefore,  have  a  bluish  tinge 
(Tf  H5),  particularly  near  the  edges  of  the  letters. 
In  the  same  way,  if  the  violet  rays  are  just  in  focus, 
reddish  or  yellowish  borders  will  encroach  upon  the 
spaces  in  question.  It  is  thus  evident  that  the  prin- 
cipal focus  of  a  lens  depends  upon  the  kind  of  light 
employed.  Green  light  may  be  taken  as  the  stan- 
dard. To  focus  for  the  green  rays,  the  distance 
of  the  lens  from  the  print  must  be  such  that  the 
black  spaces  have  very  narrow  borders  of  a  neutral 
tint ;  that  is,  one  which  inclines  neither  to  red  nor 
to  blue. 

To  obtain  the  best  results  with  the  color-method, 
a  perforated  metallic  lamp  chimney  should  be  substi- 
tuted for  the  page  of  print  (see  Exp.  42).  The 
measurements  of  distance  are  made  and  reduced 
as  in  methods  previously  described  (see  1,  2, 
and  3). 

The  student  should  make  at  least  two  determina- 
tions of  the  principal  focal  length  of  a  lens,  —  one 
by  the  ordinary  method,  the  other  by  the  indirect 
method,  (3).  The  other  methods  will  be  met  in  ex- 
periments later  on.  The  results  of  different  methods 


236  FOCAL  LENGTHS.  [Exp.  42. 

should  agree  within  limits  which  may  be  attributed 
to  errors  of  observation.1 


EXPERIMENT  XLII. 

CONJUGATE    FOCI. 


Tf  117.  Determination  of  Conjugate  Focal  Lengths 
of  Lenses.  —  A  screen,  c  (Fig.  102),  and  a  lens,  5,  are 
to  be  mounted  on  movable  supports,  as  in  Exp.  41 ; 
but  in  place  of  the  telescope  (a,  Fig.  100)  the  sup- 
port, d,  is  to  carry  a  lamp,  a,  having  a  metallic  chim- 
ney with  several  small  holes  in  it.  The  marker,  g, 


FIG.  102. 


must  be  in  line  with  the  perforations  in  the  chimney, 
not,  as  in  ^[  112,  (3)  with  the  flame,  since  the  former 
and  not  the  latter  will  be  focussed  upon  the  screen. 


1  If  in  (1)  or  (2)  the  object  is  too  near,  so  that  the  rays  from  it 
striking  the  lens  are  perceptibly  diverging,  the  distance  of  the  screen 
from  the  lens  must  evidently  be  increased  in  order  that  these  rays 
may  be  focussed  upon  it.  On  the  other  hand,  if  in  3  or  4  the 
telescope  is  focussed  upon  the  same  object,  the  distance  of  the  print 
from  the  lens  must  be  diminished  in  order  that  the  rays  which  pass 
through  the  lens  may  be  slightly  divergent ;  for  the  telescope,  being 
focussed  for  slightly  divergent  rays,  can  be  in  focus  for  no  others. 
By  averaging  a  result  obtained  by  (1)  or  (2),  with  a  result  from  (3) 
or  (4),  the  true  value  of  the  principal  focal  length  may  be  calculated, 
even  when  a  distant  view  cannot  be  obtained. 


1H7-]  CONJUGATE  FOCI.  237 

Throughout    this  experiment    the    color   method   of 
focussing  (see  ^[  116,  4)  is  to  be  used. 

(1)  The  lens  is  first  placed  in  the  middle  of  the 
bench  gi,  with  the  lamp  at  a  distance  from  it  equal  to 
twice  its  principal  focal  length,  determined  in  Experi- 
ment 41.     The  screen  is  then  moved  until  an  image 
of  the  perforations  of  the  chimney  appears  upon,it; 
the  distance  between  the  lamp  and  screen  is  then 
measured.     The  lens  will  probably  be  found  to  be 
about  half-way  between  the  lamp  and  screen ;  if  it  is 
not  exactly  in  the  middle,  it  should  be  placed  there, 
and  the  focus,  if  necessary,  readjusted  by  increasing 
or  diminishing  the  distance  of  both  the  lamp  and  the 
screen  by  an  equal  amount  in  each  case.     The  dis- 
tance of  the  screen  from  the  lamp  will  be  about  four 
times  the  principal  focal  length  of  the  lens. 

(2)  The  lamp  and  screen  are  next  separated  by  a 
distance  equal  to  about  five  times  the  principal  focal 
length  of  the  lens  ;  and  the  lens  is  placed  so  that  the 
chimney  may  be  focussed  upon  the  screen  as  before. 
Two  positions  will  be  found,  —  one  nearer  the  lamp, 
the  other  nearer  the  screen  (see  Fig.  102).     In  the 
first  position,  the  image  of  the  chimney  will  be  mag- 
nified ;  in  the  second  it  will  be  diminished  in  size 
(see  §  104).     The  second  image   vnff  be  the  more 
distinct  ;  the  first,  unless  carefully  searched  for,  may 
even  escape  detection.     The  distances  ab  and  be  are 
to  be  determined  in  each  case. 

(3)  The  lamp  and  screen  are  finally  separated  as 
far  as  possible  ;  and,  as  before,  the  lens  is  placed  so  as 
to  throw  first  a   magnified   and   second   a  reduced 


238  FOCAL   LENGTHS.  [Exp.  42. 

image  of  the  chimney  upon  the  screen.  In  both 
cases,  the  distances  ab  and  be  are  to  be  determined. 

The  distances  ab  and  be  in  each  of  the  cases  (1), 
(2),  and  (3),  are  called  conjugate  focal  lengths 
(§  103).  They  may  be  determined  by  the  readings 
of  the  markers  g,  h,  and  i.  In  (1)  the  sum  of  the 
distances  ab  and  be  is  alone  needed,  and  should  be 
confirmed  by  a  direct  measurement  with  a  metre  rod. 
If  the  markers  are  found  to  be  tolerably  accurate, 
the  readings  of  the  scale  in  (2)  and  (3)  need  not  be 
confirmed  by  direct  measurement. 

From  the  results  of  each  adjustment,  the  principal 
focal  length  of  the  lens  is  to  be  calculated  by  the  for- 
mula derived  from  that  in  §  103 :  — 

_  aJ_X_^. 
*  ~~  ab  +~bc 

The  results  should  agree  with  those  obtained  in 
Experiment  41  within  a  limit  which  may  be  attributed 
to  the  thickness  of  the  lens,  which  has  been  disre- 
garded in  the  formulae. 

The  student  should  notice  that  it  is  impossible  to 
focus  the  lamp  upon  the  screen  (1)  when  the  dis- 
tance ac  is  less  than  four  times  the  principal  focal 
length  of  the  lens,  no  matter  where  the  lens  is  placed  ; 
(2)  when  the  distance  (ab)  between  the  lamp  and  the 
lens  is  less  than  its  principal  focal  length,  no  matter 
where  the  screen  is  placed  ;  and  (3)  when  the  dis- 
tance (5c)  between  the  screen  and  the  lens  is  less 
than  its  principal  focal  length,  no  matter  where  the 
lamp  is  placed. 


IT  118.]  CONJUGATE   FOCI.  239 

It  should  also  be  noticed  that  in  (2)  and  in  (3) 
the  distances  ab  and  be,  at  which  a  magnified  image 
is  produced,  are  equal  respectively  to  the  distances 
be  and  ab,  at  which  we  obtain  an  image  reduced  in 
size ;  and  that  in  every  case  the  distance  between 
two  perforations  in  the  chimney  is  to  the  distance  be- 
tween their  respective  images  as  the  distance  of  the 
lamp  from  the  lens  is  to  that  of  the  screen  from  the 
lens  (§  104).1  It  is  hardly  necessary  to  call  atten- 
tion to  the  fact  that  all  the  images  are  inverted. 


EXPERIMENT   XLIII. 

VIRTUAL   FOCI. 

^f  118.  Real  and  Virtual  Foci  of  Mirrors.  —  Rays  of 
light  may  be  brought  to  a  focus  by  a  concave  mirror 
as  by  a  converging  lens.  If  in  Fig.  102  (^[  117)  we 
substitute  for  the  lens,  6,  a  mirror  with  its  concave 
surface  turned  towards  the  lamp,  a,  and  at  a  suffi- 
cient distance  from  it,  an  inverted  image  of  the  lamp 
will  be  formed  at  a  point  c,  between  a  and  b.  This 
image,  which  will  be  reduced  in  size,  may  be  re- 
ceived upon  a  screen,  provided  that  the  latter  is  not 
so  large  as  to  cut  off  all  light  from  the  mirror. 
Again,  if  the  screen  (<?)  is  at  a  sufficient  distance  from 
the  mirror  (5),  a  magnified  image  of  the  lamp  may 
be  thrown  upon  it  by  placing  the  lamp  at  some  point, 

1  It  is  instructive  to  prove  this  by  actual  measurement.  See  Ex- 
periment 38  in  the  Elementary  Physical  Experiments,  published  by 
Harvard  University. 


240  FOCAL   LENGTHS.  [Exp.  43. 

a,  between  b  and  c  (as  in  Fig.  104),  provided  that 
the  lamp  does  not  intercept  all  the  rays  reflected  by 
the  mirror  towards  the  screen.  In  any  case  the  real 
image  (e)  formed  by  the  mirror  is  on  the  same  side 
of  the  mirror  (6)  as  the  object  (a),  not  as  in  the  case 
of  a  lens,  on  the  opposite  side  of  it. 

The  distances  ab  and  be  are  called,  as  in  the  case 
of  a  lens,  conjugate  focal  lengths.  The  principal 
focal  length  of  a  concave  mirror  may  be  found  by 
determining  the  distance  at  which  parallel  rays  (or 
rays  from  a  sufficiently  distant  object)  are  brought 
to  a  focus,  or  by  the  formula  of  ^[  117,  applicable  to 
conjugate  focal  lengths.  These  methods  are  particu- 
larly valuable  in  the  case  of  mirrors  whose  curvature 
cannot  be  determined  by  means  of  a  spherometer 
(Experiment  21).  Evidently  the  focal  lengths  of  a 
mirror  depend  solely  on  its  curvature.  The  material 
of  which  it  is  composed  does  not,  as  in  the  case  of  a 
lens,  have  to  be  considered. 

The  images  thrown  by  a  concave  mirror  upon  a 
screen  are  instances  of  real  images.  The  image  of 
an  object  seen  in  a  plane  mirror  is  a  typical  case  of  a 
virtual  image  (§  104).  If  the  eye  is  placed  behind 
the  mirror  (where  the  image  seems  to  be)  no  light 
whatever  is  perceived.  A  thermopile  would  feel  no 
heat  there,  nor  would  photographic  paper  be  affected. 
And  yet,  as  far  as  points  in  front  of  the  mirror  are 
concerned,  the  optical,  thermal,  and  photographic 
effects  are  the  same  as  if  a  real  object  existed  behind 
the  glass. 

The  simplest  way  to  locate  a  virtual  image  is  by 


f  118] 


VIRTUAL  FOCI. 


241 


the  method  of  parallax  (^[  116,  2).  A  short  wire  is 
mounted  in  place  of  the  lamp  (#,  Fig.  102)  on  a  sup- 
port, d ;  a  longer  wire,  c,  is  attached  to  the  support/, 
and  a  piece  of  looking-glass  is  placed  between  the 
wires  on  a  support,  e,  instead  of  the  lens  b.  The 
height  of  the  wires  should  be  such  that  the  point  of 
the  long  wire,  c,  may  be  visible  above  the  image  of 
the  wire  a,  reflected  (as  in  Fig.  103)  by  the  mirror. 
As  the  eye  is  moved  from  the  farthest  left-hand  point 
(see  1  and  3,  Fig.  103)  at  which  both  wires  are 
visible,  to  the  farthest  right-hand  point  (see  2  and 
4,  Fig.  103),  both  a  and  c  (one  being  really,  the 
other  virtually,  behind  the  mirror)  will  move  from 


FIG.  103. 

the  left  of  the  mirror  to  the  right ;  but  the  one  which 
is  farthest  off  will  apparently  move  farther  than  the 
other  (see  ^  116,  2).  Thus  if,  as  in  (1)  and  (2),  the 
point  a  moves  completely  across  the  mirror,  while 
the  point  c  only  moves  part  way  across  it,  we  con- 
clude that  a  is  too  far  from  (or  c  too  near)  the  minvor, 
but  if,  as  in  (3)  and  (4),  e  moves  wholly  across  while 
a  moves  only  part  way  across,  we  conclude  that  c  is 
too  far  from  (or  a  too  near)  the  mirror.  By  adjusting 
the  distances  nb  and  be  until  no  parallax  (§  25)  is 
visible  between  a  and  c,  the  distance  of  the  virtual 
image  from  the  mirror  may  be  determined. 

16 


242  FOCAL  LENGTHS.  [Exr.  43. 

It  is  found  that  the  virtual  image  formed  by  a 
plane  mirror  is  just  as  far  behind  it  as  the  real  object 
is  in  front  of  it.1  If  a  mirror  is  slightly  convex  or 
concave,  this  will  no  longer  be  true.  A  comparison 
of  the  two  distances  ab  and  be  will  serve  therefore  to 
detect  any  curvature  in  the  surface  of  the  mirror. 

We  notice  that  virtual  images  are  never,  like  real 
images,  inverted.  When  formed  by  a  mirror  they 
are  always  behind  it.  On  the  other  hand,  we  shall 
see  that  the  virtual  focus  of  a  lens  is  always  on  the 
same  side  as  the  object. 

^[119.  Determination  of  Virtual  Focal  Lengths  of 
Lenses.  —  I.  CONVERGING  LENSES.  —  When  the  prin- 
cipal focal  length  of  a  lens  exceeds  the  limit  of  the 


FIG.  104. 

apparatus  employed,  it  can  be  determined  only  by 
means  of  virtual  foci.  Two  wires,  a  arid  c,  are 
mounted  on  sliding  supports,  as  in  Fig.  104,  on  the 
same  side  of  the  converging  lens  (5)  so  that  the  top 
of  the  farther  wire  (<?)  may  be  visible  just  above  the 
magnified  image  of  the  nearer  wire  (a)  seen  through 
the  lens.  The  wires  are  then  placed  so  that  there 
may  be  no  parallax  (§  25)  between  them  when  the 
eye  is  moved  from  side  to  side  (see  ^f  118,  Fig.  103). 
The  virtual  image  of  a  then  coincides  with  the  real 

1  This  may  be  shown  by  a  simple  geometrical  construction.     See 
Ganot's  Physics,  §  513,  Deschanel,  §  699. 


11119.]  VIRTUAL  FOCI.  243 

point,  c.     The  distances  ab  and  ac  are  then   meas- 
ured,  as  in  \  117,  and   the   principal  focal  length 
of  the  lens  is  calculated  by  the  formula  (see  §  104), 
.       ab  X  be 


II.  DIVERGING  LENSES.  —  With  diverging  lenses, 
focal  lengths  can  be  determined  only  by  the  method 
of  virtual  foci,  since  such  lenses  form  no  real  images 
(§  104).  The  method  is  essentially  the  same  as  that 
employed  with  converging  lenses  (see  I.),  except 
that  the  wire,  a,  viewed  through  the  lens,  b,  must  be 
further  off  than  the  wire,  <?,  which  is  seen  above  or 
below  it.  It  is  well  to  substitute  a  broad  netting  or 
page  of  print  for  a,  so  that  it  may  not  be  completely 
hidden  by  c. 

The  distances,  ab  and  be,  are  to  be  adjusted  so  that 
all  parallax  disappears  between  a  and  c  ;  the  virtual 
image  of  a  will  then  coincide  with  c.  The  distances 
ab  and  be  are  to  be  measured,  and  the  value  of/ 
(which  will  be  negative)  is  to  be  calculated  by  the 
same  formula  as  before.  It  may  be  noted  that  a 
virtual  image  of  distant  objects  is  formed  between  a 
diverging  lens  and  the  objects  in  question,  and  at  a 
distance  (/)  from  the  lens,  which  is  sometimes  called 
its  (virtual)  principal  focal  length. 

The  student  should  observe  that  a  converging  lens 
forms  a  virtual  image  farther  off  than  the  object 
looked  at,  while  a  diverging  lens  forms  a  virtual 
image  nearer  than  the  real  object.  Upon  this  fact 
depends  in  part  the  magnifying  power  of  a  converg- 
ing lens,  and  the  reducing  power  of  a  diverging  lens. 


244 


THE   SEXTANT. 


[Exp.  44. 


The  farther  off  an  object  is,  the  larger  must  it  be  in 
order  that  its  image  may  occupy  a  given  space  on 
the  retina ;  hence,  the  farther  off  we  think  it  is,  the 
greater  will  be  our  estimate  of  its  dimensions. 

In  the  arts,  lenses  are  often  numbered  according  to 
their  principal  focal  length.  A  No.  12  spectacle  lens 
is  generally  one  which  focusses  distant  objects  at  a 
distance  of  12  inches.  Near-sighted  or  diverging 
lenses  are  numbered  on  the  same  system.  A  No.  12 
near-sighted  lens  combined  with  a  No.  12  magnifying 
lens  should  form  a  perfectly  neutral  combination. 


EXPERIMENT  XLIV. 

THE   SEXTANT. 

^[  120.  Principle  of  the  Sextant.  —  A  sextant  may 
be  constructed,  as  in  Fig.  105,  of  two  pieces  of  look- 
ing-glass, ag  and  a;,  hinged  together  at  a  with  their 
reflecting  surfaces  in- 
ward. The  silvering  is 
removed  near  e  and  near 
t,  so  that  an  object  in 
the  direction  x  may  be 
seen  through  the  two 
glasses  ;  but  enough  sil- 
vering is  left  between  b 
andy  to  make  it  possible 
also  to  see  objects  in  the 
direction  y,  reflected  by 
the  mirror  ag  in  the  direction  hi,  then  by  aj  in  the 


FIG.  IDS. 


THEORY  OF   THE   SEXTANT.  245 

direction  ie.    The  angle,  a,  between  the  mirrors  may 
be  measured  by  a  graduated  arc,  yz. 

Let  us  first  find  the  relation  between  the  angle  d 
through  which  the  ray  y  is  bent  and  the  angle  a 
between  the  mirrors.  The  law  of  the  reflection  of 
light  (§  97)  gives  us  the  angles  b  —  j  and  g  =  h.  The 
vertical  angles  c  and  e  are  equal  by  construction,  also 
g  and/;  hence /=  h.  We  have  furthermore  in  the 
triangles  abc  and  a£/i,  — 

a  =  180°  -  b  -  c,  (1) 

a  =  180°  -b-i-h.  (2) 

Substituting  equals  for  equals,  we  have,  — 

a  =  l8Q°-j-e,  (3) 

a  =  180°  -  b  -  i  -/.  (4) 

Adding  (3)  and  (4), 

2a  =  SGQ°  -e-f-b-i-j; 
or  since  5,  i,  and"/  together  equal  180°, 

2a  =  180°  -  e  -/.  (5) 

But  from  the  triangle  def,  we  have,  — 

<Z  =  180°-e-/;  (6) 

hence,  comparing  (5)  and  (6),  we  find,— 

d  =  2a.  (7) 

We  see,  therefore,  that  when  a  ray  of  light  is 
reflected  by  two  mirrors,  the  angle  (<f)  between  its 
original  direction  (yd)  and  its  final  direction  (xd)  is 
equal  to  twice  the  angle  (a)  between  the  mirrors. 


246  THE  SEXTANT.  [Exp.  44. 

Now  let  us  suppose  that  the  plane  ayz  is  made 
vertical,  and  that  the  angle  a  is  adjusted  so  that  the 
rays  of  the  sun  *  from  the  direction  y  may  seem,  after 
being  twice  reflected,  to  come  from  the  direction  #, 
let  us  say  that  of  the  horizon ;  then  the  altitude  of 
the  sun  is  evidently  2a.  The  student  should  note 
that  two  objects  in  different  directions  may  be  visible 
simultaneously  through  a  sextant.  The  sun  may  be 
made  to  appear,  in  fact,  as  if  it  were  actually  on  the 
horizon. 

^[  121.  Description  of  an  Ordinary  Sextant.  —  We 
have  seen  how  a  sextant  may  be  constructed  out 


FIG.  106. 

of  two  mirrors  hinged  together  as  in  Fig.  105.  In 
practice  it  would  be  necessary  to  remove  most 
of  the  silvering  between  j  and  z,  since  it  would 
otherwise  interfere  with  the  ray  yd  when  the 
angle  d  is  very  small.  In  an  ordinary  sextant,  this 

1  The  mirror  should  be  smoked  near  f  and  g  before  trying  this 
experiment,  in  order  that  the  brightness  of  the  sun  may  be  sufficiently 
diminished. 


1  121.]  DESCRIPTION  OF  A   SEXTANT.  247 

portion  of  the  mirror  is  entirely  removed.  Of  the 
two  mirrors,  az  and  ag,  there  remain  in  fact  only  the 
small  portions,  bj  and  hgy  represented  respectively  by 
a  and  b  in  Fig.  106,  or  by  ac  and  df  in  Fig.  107. 
The  mirror  a  (Fig.  106)  is  fixed  in  position,  and  b  is 
pivoted  at  its  centre  instead  of  an  axis  (as  in  Fig. 
105)  where  the  planes  of  the  two  mirrors  intersect.1 
The  angle  between  these  planes  is  moreover  meas- 
ured, not  by  an  arc  (zy,  Fig.  105)  included  by  the 
angle  (a),  but  by  an  arc  g  (Fig.  106)  situated  in 
quite  a  different  part  of  the  instrument.  On  this  arc 
a  vernier  (K)  connected  by  a  movable  arm  with  the 
mirror  (6)  serves  to  indicate  the  angles  through 
which  the  mirror  (5)  is  turned. 

A  tube  or  telescope,  c  (Fig.  106),  permanently 
pointed  toward  the  fixed  mirror  (a)  serves  princi- 
pally as  a  guide  for  the  eye.  There  is  also,  in  most 
sextants,  a  set  of  dark  glasses,  df,  which  may  be  so 
placed  as  to  diminish  the  light  of  the  sun  when 
looked  at  directly  through  the  unsilvered  part  of 
the  fixed  mirror,  a ;  there  is  also  a  set  of  dark  glasses 
at  e  (not  shown  in  the  figure)  to  cut  off  excessive 
light  reflected  by  the  revolving  mirror,  b.  A  magni- 
fying glass  (/)  is  used  for  reading  the  vernier  (h). 
The  vernier  is  clamped  by  a  thumb-screw  (/),  and 
slow  motion  is  produced  (only  when  clamped)  by  the 
tangent  screw  (i).  There  is  also  a  screw  (I)  by  which 

1  The  fixed  mirror,  a,  is  called  the  "horizon-glass,"  because  in 
nautical  observations  the  horizon  is  usually  seen  through  it;  the 
revolving  mirror,  b,  is  called  the  "  index-glass  "  because  it  carries  the 
index.  See  Glazebrook  and  Shaw's  Practical  Physics,  §  48. 


248  THE   SEXTANT.  [Exp.  44. 

the  tube  or  telescope  (c)  may  be  either  raised  so  as  to 
come  opposite  the  upper  portion  of  the  mirror,  a, 
which  is  unsilvered,  or  lowered  so  as  to  be  opposite 
the  silvered  portion.  By  this  means,  the  relative 
brightness  of  the  direct  and  doubly  reflected  images 
may  be  varied  at  pleasure.  The  handle  k  is  of  use 
especially  in  nautical  observations. 

^f  122.  Adjustments  and  Reading  of  a  Sextant.  — 
In  order  that  a  sextant  may  give  accurate  readings, 
certain  conditions  must  be  fulfilled. 

(1)  The  tube  or  telescope,  c,  must  be  parallel  to 
the  plane  of  the  graduated  arc  ;  for  in  demonstrating 
the   relation    between   the    angle    (xdy,   Fig.    105) 
through  which  a  ray  of  light  is  bent  and  the  angle 
(a)  between  the  mirrors,  we  have  assumed  that  the 
whole  figure  lies  in  one  plane.     This   condition  is 
fulfilled  if  a  distant  object,  visible  through  the  tube 
or  telescope  (c)  in  the  middle  of  the  field,  appears, 
when  sighted,  to  be  in  the  same  plane  as  the  gradu- 
ated arc.     If  this  condition  is  not  fulfilled,  the  posi- 
tion of  the  tube  or  telescope  must  be  altered  by  an 
instrument-maker,  so  that  the  line  of  sight  may  be 
parallel  to  the  plane  of  the  graduated  arc. 

(2)  The  pivot  on  which  the  mirror  (6,  Fig.  106) 
rotates  must  be  perpendicular  to  the  plane  of  the 
graduated   arc.     This   condition   is   fulfilled    if  the 
movable  arm  can  be  turned  from  one  end  of  the  arc 
to   the   other  without  either  leaving   it  or  binding 
against  it.     If  it  is  not  fulfilled,  the  sextant  should 
be  discarded. 

(3)  The  revolving  mirror  should  be  perpendicular 


T  122.]  ADJUSTMENTS  OF  A   SEXTANT.  249 

to  the  plane  of  the  graduated  arc.  This  condition  is 
fulfilled  if  the  reflection  of  the  arc  in  the  mirror 
seems  to  be  a  continuation  of  this  arc.  If  the  re- 
flected portion  seems  to  slope  upward  or  downward, 
the  mirror  leans  forward  or  backward.  The  adjust- 
ment of  the  revolving  mirror  should  not  be  attempted 
by  the  student,  but  should  be  left  to  the  instrument- 
maker. 

(4)  The  fixed  mirror  should  be  perpendicular  to 
the  plane  of  the  graduated  arc.     This  condition  is 
fulfilled  if,  after  the  revolving  mirror  has  been  prop- 
erly adjusted,  the  sextant  can  be  set  so  as  to  give  a 
single  image  of  distant  objects ;  for  the  fixed  mirror 
is  then  parallel  to  the  revolving  mirror,  and  hence 
perpendicular  to  the  arc.     The  reading  of  the  sex- 
tant when  so  set  is  called  its  zero-reading  (see  ^[  123). 
If  no  such  setting  can  be  made,  the   fixed   mirror 
should  be  tipped  a  little  forward  or  backward  by 
turning  one  of  the  screws  which  hold  it  in  place. 
This  adjustment  should  be  attempted  only  by  per- 
sons who  have  acquired  some  skill  in  the  use  of  a 
sextant. 

(5)  The  fixed  mirror  should  be  nearly  parallel  to 
the  revolving  mirror  when  the  index  attached  to  the 
latter  points  to  the  zero  of  the  graduated  arc.     This 
is  the  case  if  the  sextant  gives  only  a  single  image  of 
distant  objects   when   set   as   stated.      If  a   double 
image   is   seen,  one    of  the  two  mirrors  should  be 
rotated  without  disturbing  the  setting.     A  screw  is 
usually  provided  for  rotating  the  fixed  mirror  through 
a  small  angle.     There  is  danger  in  so  doing  that  the 


250  THE    SEXTANT.  [Exp.  44. 

last  adjustment  (1)  may  be  disturbed.  If  it  is,  it 
must  be  repeated.  The  student  is  advised  to  omit 
the  5th  adjustment  altogether,  since  a  slight  error 
in  it  may  cause  a  little  inconvenience  in  allowing 
for  zero  error,  but  will  not  affect  the  accuracy  of 
results. 

(6)  The  arc  and  vernier  must  each  be  uniformly 
graduated.     The  uniformity  of  the  arc  may  be  tested 
(as  in  ^[  48  d)  by  means  of  the  vernier.     If  the  latter 
subtends,  for  instance,  119  divisions  in  all  parts  of 
the  arc,  these  divisions  must  have  the  same  length. 
If  the  coincidences  on  the  vernier  follow  in  regular 
succession  as  the  tangent  screw  (i)  is  slowly  revolved, 
we  may  infer  uniformity  both  in  the  main  scale  and 
in  the  vernier. 

(7)  The  value  of  the  main-scale  and  vernier  di- 
visions  must  be   known.     An   accurate   method  of 
correcting  the  main  scale  will   be  considered  (inci- 
dentally) in  Experiment  45.     To  decide  whether  the 
divisions,  of  which  every  tenth  one  is  usually  num- 
bered,  are   intended   to   be    degrees,   or  only   half- 
degrees,  so  as  to  represent  the  number  of  degrees 
through  which  a  ray  of  light  is  bent  (see  ^[  120,  for- 
mula 7),  a  rough  test  will  be  sufficient.     Thus  if  a 
string  reaching  from  the  pivot  to  the  graduated  arc 
also  reaches  from  0  to  120  on  the  arc,  we  may  infer 
that  the  divisions  are  half-degrees.     Ity  calling  them 
degrees  we  shall  avoid  the  labor  of  doubling  each 
reading  of  the  sextant  when  measuring  the  angle 
through  which  a  ray  is  bent  by  reflection  in  the  two 
mirrors. 


1  122.]  READINGS  OF   SEXTANTS.  251 

The  divisions  which  represent  degrees  are  divided 
in  different  instruments  into  two,  three,  four,  six,  and 
even  twelve  parts.  The  number  of  minutes  corre- 
sponding to  each  part  is  easily  calculated.  The 
vernier  usually  contains  lines  of  different  lengths. 
There  are  as  many  of  the  longest  lines  as  there  are 
minutes  in  the  smallest  main-scale  division.  These 
lines  are  not  usually  so  close  together  as  the  main- 
scale  divisions,  but  by  paying  attention  simply  to  the 
number  of  the  long  line  which  coincides  most  nearly 
with  some  main-scale  division,  we  find  the  number  of 
minutes  to  be  added  to  the  reading  of  the  main  scale 
(see  §  40).  Between  the  long  lines,  shorter  lines  are 
frequently  placed,  to  represent  fractions  of  a  minute. 
Since  a  setting  made  by  the  eye,  unaided  by  the 
telescope,  is  hardly  accurate  to  a  minute,1  the  stu- 
dent is  advised  to  disregard  these  lines  until  he  has 
mastered  the  reading  of  the  sextant  to  degrees  and 
minutes. 

In  angular  as  in  linear  measure,  there  is  danger  of 
making  a  mistake  of  a  whole  main-scale  division 
(Tf  50,  II.).  If  the  reading  of  the  main  scale  is  thought 
to  be  about  x°,  and  the  vernier  shows  it  to  be  a 
whole  number  plus  1',  we  record  this  reading  as 
x°  1' ;  but  if  the  vernier  indicates  a  whole  number 
plus  59',  we  record  the  reading,  not  as  x°  59',  but 
(x  —  1)°59'. 

1  A  man  four  miles  off  would  subtend  an  angle  of  about  one  minute. 
A  minute  corresponds  to  a  distance  of  less  than  one  three-hundredth  of 
an  inch  on  a  piece  of  paper  held  at  the  ordinary  distance  (10  inches) 
from  the  eye. 


252  THE   SEXTANT.  ,        [Exp.  44. 

The  first  degree-mark  below  zero  is  counted  as 
minus  one  ;  the  second,  minus  two,  etc.  The  number 
of  minutes  is  always  positive,  since  the  vernier  is 
made  to  read  this  way.  To  avoid  confusion,  the 
negative  sign  is  written  over  the  number  of  degrees, 
which  it  alone  affects  (see  ^[  50,  I.).  Thus,  a  nega- 
tive angle  of  —  21'  would  be  recorded  1°  39'. 

^[  123.  Determination  of  the  Zero-Reading  of  a  Sex- 
tant. —  After  a  sextant  has  been  adjusted  as  accu- 
rately as  possible  (see  ^[  122),  its  zero-reading  must 
be  determined.  The  index  is  first  set  at  the  zero  of 
the  main  scale  (as  in  ^[  122,  5),  the  dark  glasses  are 
pushed  out  of  the  way,  and  the  tube  or  telescope  (c)  is 
directed  toward  some  distant  object,  —  the  smaller 
and  brighter  the  better.  A  star  is  universally  con- 
ceded to  be  the  best  object,  but  a  distant  electric  arc- 
light  will  do.  In  the  day-time,  a  church  spire  or  the 
top  of  a  flag-pole  may  answer.  At  sea  the  hori- 
zon line  is  frequently  employed ;  in  this  case  the 
plane  of  the  sextant  must  be  vertical.  The  angle 
between  the  mirrors  should  be  so  slight  that  the  di- 
rect and  doubly  reflected  images  of  the  given  object 
may  at  least  be  included  in  the  same  field  of  view. 
These  images  are  then  made  to  coincide  by  turning 
the  tangent-screw  (z,  Fig.  106).  Finally,  the  reading 
of  the  sextant  is  taken.  This  is  called  its  zero-read- 
ing, because  it  corresponds  to  an  angle,  zero,  between 
the  direct  and  doubly-reflected  rays. 

It  is  easy  to  show  that  the  fixed  and  revolving 
mirrors  must  be  parallel  when  these  rays  (yl  and  xg, 
Fig.  107)  are  parallel ;  for  the  alternate  interior 


IT  124.]  MEASUREMENT  OF  ANGLES.  253 

angles  b  and  e  are  equal  by  construction,  hence  their 
supplements,  a-{-c  and  d -{-/must  be  equal.  Now, 
the  law  of  the  reflection  of  light  <A 

(§    97)    gives    a  =  c,   and    d=f;  ~3% 

hence,  a  being  half  of  a  -f-  c  must  be         ^--? 
equal  to  d,  which  is  half  of  d  -f  / 
Since  c  and  d  are  alternate  interior 
angles  formed   by   the  intersection   of  be   with  the 
mirrors  ac  and  df^  these  mirrors  must  be  parallel. 
Conversely,  if  the  mirrors  are  parallel,  the  direct  and 
doubly-reflected  rays  must  be  parallel. 

In  order  that  the  rays  yb  and  xg  may  be  sensibly 
parallel,  let  us  say  within  one  minute  (!')  of  angle, 
the  object  from  which  they  come  must  be  3,438  times 
as  far  off  from  the  sextant  as  these  rays  are  from 
each  other.  Since  the  perpendicular  distance,  bg^  is 
generally  less  than  a  twelfth  of  a  metre,  it  may  be 
safe  to  employ  any  object  more  than  300  metres  off 
for  the  determination  of  the  zero-reading  of  a  sextant 
with  the  unaided  eye.  To  obtain  results  accurate 
to  half  a  minute,  the  minimum  distance  must  be 
doubled ;  for  accuracy  within  10"  of  angle  the  object 
should  be  at  least  1,800  metres,  or  more  than  a  mile 
away.  For  such  results,  a  telescope  (c,  Fig.  106) 
must  be  employed. 

<[f  124.  Determination  of  Small  Angular  Magnitudes 
by  means  of  a  Sextant.  —  I.  A  sextant  is  to  be  set  at 
or  near  its  zero-reading ;  then  turned  so  that  the 
telescope  (c,  Fig.  106)  may  point  directly  toward  the 
sun.  The  sextant  is  to  be  held  so  that  its  graduated 
arc  may  be  in  a  vertical  plane,  below  the  revolving 


254  THE   SEXTANT.  [Exr.  44. 

mirror  (5).  A  sufficient  number  of  dark  glasses 
must  be  interposed  in  the  paths  of  both  the  direct 
and  the  reflected  rays.  It  is  well  to  select  these 
glasses  so  that  the  two  images  of 
the  sun  may  differ  in  color,  and 
^  ^  thus  be  easily  distinguished.  The 
tangent  screw  (i,  Fig.  106)  is  to 
be  turned  until  the  doubly-re- 
flected image  (R.  I.,  Fig.  108) 
appears  to  be  tangent  to  the  di- 

in  (1).  The  vernier  is  then  read.  Next,  by  turning 
the  tangent  screw  the  other  way,  the  reflected  image 
(R.  I.)  is  made  to  move  completely  through  the  di- 
rect image,  until  it  is  tangent  to,  and  above  it,  as  in 
(2).  The  vernier  is  again  read. 

The  first  reading  should  be  positive,  the  second 
negative.  The  average  of  the  two  should  be  found 
and  compared  with  the  zero-reading  previously  de- 
termined, with  which  it  should  agree.  If  the  differ- 
ence exceeds  1',  the  measurements  in  *|J^[  123  and  124 
should  be  repeated. 

The  second  reading  is  now  to  be  subtracted  (alge- 
braically) from  the  first.  The  difference,  divided  by 
2,  is  evidently  the  angular  diameter  of  the  sun.  The 
semi-diameter,  which  is  quoted  in  all  nautical  alma- 
nacs, varies  from  month  to  month,  according  to  the 
earth's  distance  from  the  sun.  Its  mean  value  is  not 
far  from  16'. 

II.  The  sextant  may  also  be  used  for  the  determi- 
nation of  the  angular  diameter  of  small  terrestrial 


f  125.]  REFLECTION  OF  LIGHT.  255 

objects.  The  plane  of  the  graduated  arc  must  be 
held  in  all  cases  so  as  to  be  parallel  to  the  diameter 
which  it  is  desired  to  measure.  The  object  should 
be  so  small  that  a  negative  *  as  well  as  a  positive 
reading  may  be  obtained,  as  in  the  case  of  the  sun. 
The  average  of  the  two  readings  should  agree  with  a 
zero-reading  obtained  from  the  same  object,  or  from 
one  at  an  equal  distance.  The  difference  between 
the  two  readings  is  not  affected  by  parallax,  since  the 
error  in  both  readings  is  the  same.  This  difference, 
divided  by  2,  is  therefore  the  angular  diameter  of  the 
object  in  question  as  seen  from  the  pivot  of  the  re- 
volving mirror.  The  position  of  this  pivot  should  be 
noted,  or  the  results  will  have  no  meaning.  It  is 
well  for  the  student  to  measure  either  the  actual  di- 
ameter or  the  distance  of  the  object  in  question,  still 
better,  both  of  these  quantities  ;  for  though  either 
may  be  calculated  from  the  other,  the  two  together 
give  him  the  means  of  testing  his  inferences  as  to  the 
manner  in  which  his  sextant  should  be  read  and  an 
opportunity  of  confirming  his  results. 


EXPERIMENT  XLV. 

PRISM   ANGLES. 

Tf  125.  Determination  of  the  Angles  of  a  Prism. — I.  A 
small  prism  (afo,  Fig.  109)  is  fastened  to  the  revolv- 

1  A  sextant  should  be  capable  of  giving  negative  readings  down  to 
3,  4,  or  even  6  degrees. 


256  THE   SEXTANT.  [Exp.  45. 

ing  mirror  (be)  of  a  sextant  with  its  axis  parallel,  as 
nearly  as  possible,  to  that   about  which  the  mirror 
turns.1   The  mirror  is  then 
rotated  so  that  the  direct 
image  of  a  distant  object, 
o^,-''.-'  seen  in   the  direction  ed, 
"\":" ~fl     '    T    may  coincide  with  the  im- 

, ...-u.., j4"  * 

'  age  of  the  same  object  re- 

flected first  by  the  face  of 

the  prism  (ac)  then  by  the  fixed  mirror  (c?/).  If  the 
two  images  cannot  be  made  to  coincide,  the  face  ac 
is  probably  not  parallel  to  the  axis  of  the  mirror,  and 
must  be  made  so  by  tilting  the  prism  either  from  b  to 
c,  or  from  c  to  5,  without  separating  the  two  faces,  be, 
of  the  prism  and  of  the  mirror.  When  parallelism  is 
.established,  an  exact  coincidence  of  the  images  may 
be  brought  about.  A  reading  of  the  sextant  is  then 
made.  This  serves  to  determine  the  prism  angle  c. 
In  the  same  way  the  other  two  angles  are  deter- 
mined.2 

Subtracting  from  each  reading  of  the  sextant  its 
zero-reading,  determined  as  in  ^[  123,  we  have  the 
indicated  value  of  the  angle  corresponding  to  c  (or 
acb')  in  the  figure ;  for  it  is  evident  that  the  mirror  cb 
in  rotating  from  its  zero  position,  <?a,  to  the  position 

1  The  plane  of  the  fare,  ac,  should  strictly  pass  through  the  axis 
of  the  mirror,  to  avoid  errors  of  parallax.     In  practice,  however,  it  is 
more  convenient  to  mount  the  prism  as  in  Fig.  109. 

2  To  measure  the  three  angles  of  a  prism,  one  of  which  must  be 
at  least  60°,  a  sextant  reading  to  120°  will  be  required.     "  Octants  " 
are  sometimes  graduated  to  120° ;  but  do  not  read  generally  to  more 
than  100°,  on  account  of  the  space  occupied  by  the  vernier. 


f  126.J  REFRACTION  OF  LIGHT.  257 

c6,  turns  through  the  angle  acb.  What  we  want, 
however,  is  the  actual  value  of  this  angle,  not  the 
deviation  of  a  ray  of  light  striking  the  revolving  mir- 
ror, which  plays  no  part  in  the  measurement.  If, 
therefore,  the  sextant  is  found,  as  in  If  122,  7,  to  be 
graduated  in  half-degrees,  half-minutes,  etc.,  the  in- 
dicated value  of  the  angle  must  be  halved  in  order 
to  find  the  real  value  of  acb. 

The  sum  of  the  three  prism  angles  should  be  180°. 
A  discrepancy  of  one  or  two  minutes  may  be  attrib- 
uted (1)  to  errors  of  observation,  (2)  to  pyramidal 
convergence  of  the  sides  of  the  prism,  and  (3)  to 
errors  in  the  adjustment  or  graduation  of  the  sextant. 
If  the  measurements  are  several  minutes  in  error, 
they  should,  be  repeated.  If  the  same  result  is  ob- 
tained, the  parallelism  of  the  prism  faces  should  next 
be  tested  with  a  three-pointed  caliper.  With  a  per- 
fect equilateral  prism,  we  have  evidently  the  means 
of  detecting  any  error  in  the  location  of  the  60°  mark 
(or  that  numbered  120°). 

II.  Instead  of  a  sextant,  a  spectrometer  may  be 
used,  as  will  be  explained  in  <[[  126. 


EXPERIMENT   XLVL 

ANGLES   OP   REFRACTION. 

^[  126.  The  Spectrometer.  — A  spectrometer  consists 
essentially  of  two  telescopes  (ab  and  fg,  Fig.  110) 
capable  of  revolving  about  the  centre  of  a  graduated 
17 


258  THE   SPECTROMETER.  [Exp.  46 

circle  (ede).  The  eye  piece  of  the  first  telescope 
is  generally  removed,  and  a 
narrow  slit  (a,  Fig.  110)  is 
usually  substituted  for  the 
cross-hairs  (c,  Fig.  101,  ^[  116). 
FIG.  no.  This  slit  is  always  at  right- 

angles  with  the  graduated  circle,  and  at  a  distance 
from  the  lens,  b,  equal  to  its  principal  focal  length  ; 
so  that  the  rays  from  it  may  be  rendered  parallel  by 
this  lens  (see  ^[  116,  3).  The  combination  (a&)  is 
called  the  "  collimator  "  of  the  spectrometer.  The 
telescope/^  is  focussed  for  parallel  rays  (^[  116,  3),  and 
carries  an  index  with  a  vernier,  by  which  its  position 
on  the  graduated  circle  may  be  accurately  determined. 
A  zero-reading  can  be  found  by  pointing  the  tele- 
scope toward  the  collimator  as  in  Fig.  110,  and  ad- 
justing it  so  that  the  image  of  the  slit,  a,  may  be 
visible  in  the  centre  of  the  field  of  view,  which  is 
determined  by  the  intersection  of  cross-hairs. 

Let  us  now  suppose  that  it  is  desired  to  measure 
the  angles  of  a  prism.     The  latter  is  mounted  as  in 
Fig.  Ill  (cde~),  so  that  the  face 
ce  may  reflect  part  of  the  light 
from  ab  in  the  direction  fg,  and 
so  that  at  the  same  time  the  face 
cd  may  reflect   light   in  the  di- 
rection f'y'.      The    telescope    is 
then  set  so  as  to  receive  first  one, 
then  the  other  of  the  images  of  the 
slit,  thus  formed,  in  the  middle  of  its  field  of  view, 
and  in  each  case  a  reading  of  the  vernier  is  made. 


U  127.]  REFRACTION   OF   LIGHT.  259 

Let  us  suppose  that  the  collimator  is  perma- 
nently set  at  0°  (or  360°)  of  the  circle  ;  that  fg  is  at 
x°  &nd/y  at  y°  of  the  circle.  A  radius  of  the  circle 
perpendicular  to  ce  would  halve  the  angle  a;,  on  ac- 
count of  the  law  of  reflection  (§  97)  ;  and  hence 
would  meet  the  circle  at  a  point  \  x°.  In  the  same 
way  a  radius  perpendicular  to  cd  would  meet  the 
circle  half-way  between  y°  and  360° ;  or  at  J  y°  + 
180°  ;  hence  if  prolonged  backward  it  would  meet  the 
circle  at  \  y°.  Now,  the  angle  between  two  surfaces 
may  be  measured  by  the  angle  between  two  lines 
perpendicular  to  them  ;  hence  the  difference  be- 
tween \  x°  and  \  y°  measures  the  prism  angle  dee. 
In  other  words,  the  angle  between  two  faces  of  a 
prism  is  equal  to  half  the  angle  between  the  two 
directions  in  which  they  reflect  parallel  rays  of  light. 
(Compare  ^[  120,  7.) 

The  most  important  adjustments  of  a  spectrometer 
are  the  accurate  levelling  and  focussing  of  the  tele- 
scope and  collimator  for  parallel  rays  (see  ^[  116,  3). 
The  faces  of  the  prism  must  be  made  perpendicular 
to  the  plane  of  the  graduated  circle  as  in  ^[  125.  An 
instrument  especially  adapted  to  measure  the  angle 
between  two  reflecting  surfaces  is  sometimes  called 
a  goniometer. 

^[  127.   Determination  of  Angles 
of  Refraction.  —  I.  The  telescope    c 
O#),  and  collimator  (05)   of  a 
spectrometer  are  slightly  inclined 
as  in  Fig.  112,  so  that  a  spectrum  (^[  128)  of  the  slit, 
a,  may  be  formed  in  the  telescope  by  a  prism  dee, 


260  THE   SPECTROMETER.  [Exp.  46. 


the  angles  of  which  have  been  determined  (^[^[  125, 
and  126).  The  angle  c,  causing  the  refraction,  should 
be  placed  symmetrically  with  respect  to  the  telescope 
and  colliinator.  If  dee  is  an  equilateral  prism,  an 
image  of  the  slit  may  also  be  formed  in  the  telescope 
by  reflection  from  the  face  de.  It  is  found  that  when 
the  faces  cd  and  ce  are  as  stated  equally  inclined  to 
the  rays  ab  and  fg,  the  angle  between  these  rays 
reaches  a  minimum. 

To  make  sure  that  this  position  has  been  approxi- 
mately found,  the  prism  should  be  rotated  a  little. 
The  violet  of  the  spectrum  should  be  replaced  by 
blue,  green,  yellow,  and  red,  until  finally  the  spec- 
trum disappears  altogether.  It  should  make  no  dif- 
ference whether  the  prism  is  turned  to  the  right  or  to 
the  left.  If  the  spectrum  moves  in  opposite  directions 
when  the  prism  is  turned  in  opposite  directions,  the 
desired  position  has  not  been  found.  In  this  case 
the  rotation  should  be  continued  in  one  direction  or 
the  other  until  the  spectrum  seems  to  come  to  a 
standstill.  The  prism  is  then  very  nearly  in  its  "  po- 
sition of  minimum  deviation." 

The  slit  should  now  be  illuminated  with  light  from 
a  sodium  flame,1  the  reflected  image  if  necessary  cut 
off,  and  the  telescope  roughly  set  on  the  yellow  re- 
fracted image  of  the  slit.  Then  the  prism  is  turned 
slightly  so  that  this  image  may  move  as  far  as  possible 
towards  the  red  (or  less  refrangible)  end  of  the  spec- 
trum. The  telescope  is  again  set  on  the  yellow  image 

i  A  common  Bunsen  hurner  beneath  a  netting  of  fine  iron  wire 
sprinkled  with  nitrate  of  soda  furnishes  an  excellent  "  sodium  flame." 


If  127.]  REFRACTION  OF  LIGHT.  261 

more  carefully  than  before,  and  the  prism  turned  first 
to  the  right,  then  to  the  left,  so  as  to  find  if  possible 
a  position  in  which  the  yellow  image  is  even  less 
refracted  than  before.  Thus  by  successive  approxi- 
mations, the  telescope  may  finally  be  set  upon  an 
image  of  the  slit  formed  by  the  prism  in  its  position 
of  minimum  deviation. 

Subtracting  the  zero-reading  (^[  126)  of  the  tele- 
scope from  its  reading  when  set  upon  the  refracted 
image,  we  have  finally  the  angle  of  minimum  devia- 
tion in  question ;  that  is,  the  least  angle  through 
which  sodium  light  may  be  bent  in  passing  through 
the  prism  angle,  dee,  in  the  figure. 

The  relation  between  angles  of  refraction  and  in- 
dices of  refraction  is  considered  in  §  102. 

In  repeating  the  experiment,  the  prism  should  be 
rotated  through  180",  so  that  the  rays  would  be  bent 
upward  instead  of  downward  as  in  the  figure.  If  the 
position  of  the  collimator  is  unchanged,  any  error 
in  the  zero-reading  may  be  eliminated  (see  §  44)  by 
averaging  the  result  with  that  previously  obtained. 

II.  Instead  of  the  spectrometer  a  sextant  may  be 
employed  for  the  determina- 
tion of  angles  of  refraction. 
The  prism  is  to  be  mounted 
as  in  Fig.  113,  so  that  a  ray                    .-•'' 
of  light  from  a  distant  point    jLs. .fTT.     7 

may   be    refracted    by    the 

FIG.  113. 

prism  angle  c,  previously  de- 
termined fl[  125),  then  reflected  by  the  revolving  mirror 
d,  and  by  the  fixed  mirror  e  into  the  telescope,/,  where 


262  SPECTRA.  [Exp.  46. 

it  is  made  to  coincide  with  the  direct  ray,  ef,  from  the 
same  object.  To  obtain  accurate  results,  monochro- 
matic light  should  be  employed  ;  but  a  mean  index  of 
refraction  may  be  found  by  making  the  direct  image  of 
a  flame  coincide  with  the  yellow  or  green  of  its  spec- 
trum (§  128).  The  prism  must  be  placed  by  trial  in 
the  position  of  minimum  deviation  as  with  the  spec- 
trometer. The  angle  of  deviation,  being  twice  the 
angle  between  the  mirrors,  is  indicated  directly  by 
the  reading  of  the  sextant,  after  the  zero-reading 
has  been  subtracted. 

The  use  of  the  sextant  for  the  determination  of 
angles  of  refraction  is  recommended  only  to  those 
who  have  some  skill  in  physical  manipulation.  For 
this  reason  a  detailed  description  of  the  experiment 
has  not  been  given. 

^[  128.  Spectra  formed  by  the  Dispersion  of  Light. 
—  The  rays  of  light  from  a  sodium  flame,  when  bent 
(as  in  ^[  127)  by  a  prism,  produce,  with  ordinary  ap- 
paratus, a  single  yellow  image  of  the  flame.  A  flame 
colored  with  lithium  gives  similarly  a  red  image,  and 
one  colored  with  thallium  a  green  image.  These 
images  are  not,  however,  in  the  same  direction  from 
the  observer,  owing  to  the  fact  that  rays  of  different 
hues  are  unequally  bent  by  a  prism.  Indeed,  if  a 
flame  be  colored  by  a  mixture  containing  certain  pro- 
portions of  lithium,  sodium,  and  thallium,  three  images 
of  the  flame  —  one  red,  one  yellow,  and  one  green  — 
may  be  seen  side  by  side,  distinctly  separated  by 
dark  spaces  between  them.  Many  substances,  even 
when  chemically  pure,  cause  under  the  same  circum- 


IT  128.]  DISPERSION   OF   LIGHT.  263 

stances  several  distinct  images  of  a  flame  to  be  pro- 
duced. Each  of  these  images  differs  in  hue  from  the 
rest.  The  images  may  be  more  or  less  bright  and 
more  or  less  widely  separated.  Together  they  con- 
stitute what  is  called  the  spectrum  of  the  substance 
producing  them.  When,  as  in  a  common  gas-flame, 
light  of  every  hue  is  represented,  an  indefinite  num- 
ber of  images  are  formed,  and  these  necessarily  over- 
lap one  another.  The  result  is  called  a  continuous 
spectrum. 

An  instrument  intended  simply  to  examine  spectra 
with  a  view  to  observing  the  number  of  images  pres- 
ent, is  called  a  spectroscope.  An  instrument  like 
that  described  in  ^[  126,  especially  adapted  to  the 
determination  of  angles  of  refraction,  through  set- 
tings made  upon  the  differently  colored  images  in  a 
spectrum,  is  properly  called  a  spectrometer. 

Those  substances  which  bend  light  the  most  usu- 
ally produce  the  greatest  separation  or  "  dispersion  " 
of  rays  of  different  colors.  There  is,  however,  no 
definite  proportion  between  the  effects  of  refraction 
and  dispersion.  Thus  an  equilateral  prism  of  crown 
glass  which  bends  rays  of  light  about  40°,  separates 
the  extreme  red  and  violet  rays  by  about  4° ;  while  a 
prism  of  flint  glass,  producing  nearly  double  the  dis- 
persion, bends  rays  less  than  50°. 

To  determine  the  dispersive  power  of  a  given  sub- 
stance, two  indices  of  refraction  are  generally  found 
(see  §  102),  one  with  red  light,  the  other  with  violet 
light.  The  red  light  selected  is  of  a  peculiar  wave- 
length (§  98),  namely,  .00007604  cm.,  being  that  which 


264 


WAVE-LENGTHS. 


[Exp.  47. 


causes  the  line  A  in  the  solar  spectrum.  The  violet 
light  has  similarly  a  wave-length  .00003933  cm.,  cor- 
responding to  the  line  H2  of  the  solar  spectrum. 
The  difference  between  the  indices  of  refraction  of  a 
given  substance  for  these  two  rays  is  sometimes 
called  the  "  index  of  dispersion  "  of  the  substance  in 
uuestion. 


EXPERIMENT   XL VII. 

WAVE-LENGTHS. 

1[  129.    Theory  of  the  Diffraction  Grating.  —  When  a 
distant  candle  is  looked  at  through  a  linen  handker- 
chief, or  through  any  fine  network,  several  images  .of 
the    candle    are    usually 
seen  (Fig.  114).     These 
are  not,  however,  as  one 
is  at  first  apt  to  suppose, 
simply  so  many  views  of 
the    candle    through    the 
meshes   of  the    handker- 
chief; for  each  image  rep- 
resents the  whole  candle, 
and  the  distance  between 
the    images   is   not    only 

disproportionate  to  the  size  of  the  meshes,  but  actu- 
ally increases  as  the  meshes  become  smaller.  It  is, 
moreover,  unaffected  by  the  distance  of  the  handker- 
chief from  the  eye.  The  phenomenon  is  an  example 
of  diffraction  (§§  100,  101),  and  depends  upon  a  re- 


129.] 


DIFFRACTION  OF  LIGHT. 


265 


latioii  between  the  length  of  the  waves  of  light  and 
the  distance  between  the  threads. 

The  central  image  (a,  Fig.  114)  is  the  direct  image 
of  the  candle.  It  may  be  distinguished  from  the 
side  images,  a  and  a",  for  instance,  both  by  its 
greater  distinctness  and  by  the  absence  of  color. 


FIG.  115. 


The  side  images  will  be  found  tinged  with  blue  on 
the  side  toward  a,  and  with  red  on  the  outer  side. 
This  is  due  to  the  fact  that  different  colors  are  un- 
equally bent  by  diffraction.  Each  of  the  side  images 
is  in  fact  a  "spectrum''  of  the  candle.  It  is  inter- 
esting to  place  two  caudles  at  points  corresponding 


FIG.  116. 

to  a  and  I  (Fig.  115)  at  such  a  distance  that  when 
the  candles  are  viewed  through  a  network  de,  the 
side  image  at  the  right  of  a  may  coalesce  with  the 
side  image  at  the  left  of  b,  so  as  to  form  a  single 
image  at  the  point  c  similar  to  that  represented  in 


266  WAVE-LENGTHS.  [Exp.  47. 

Fig.  116.  If  o  is  one  of  the  threads,  d  and  e  the  spaces 
between  it  arid  the  two  parallel  threads  on  either  side 
of  it,  then  drawing  ad,  ao,  ae,  cd,  co,  ce,  etc.,  also  df 
perpendicular  to  ao,  we  have  (since  ao  practically  bi- 
sects the  angle  a)  ad  =  af.  The  path  ae  is  accord- 
ingly longer  than  ad  by  the  distance  ef,  which  must 
therefore  be  the  length  of  a  wave  of  light  (§  101), 
since  the  rays  do  not  interfere.  Now,  by  similar 
triangles,  we  have, 

ef:  de  : :  ac:  ao\  I. 

hence,  if  we  know  the  distance,  de,  between  the 
threads,  the  distance,  ao,  between  the  handkerchief 
and  one  of  the  candles,  and  the  distance,  ab,  between 
the  candles,  so  that  by  halving  the  latter  the  dis- 
tance of  the  side  image  (ac)  may  be  found,  we  may 
calculate  the  average  length  (Z)  of  a  wave  of  light 

by  the  formula, 

de^ac 

ao 

The  student  may  himself  estimate  wave-lengths  in 
this  way.  For  a  human  eye  in  its  normal  condition 
(see  1T 115)  the  average  wave-length  in  a  candle-flame 
has  been  found  to  be  about  60  millionths  of  a  cen- 
timetre. We  may  make  use  of  this  fact  to  estimate 
the  distance  (d)  between  the  threads  of  the  hand- 
kerchief by  the  formula  derived  from  I., 

d  =  .00006  x  -•  III. 

ac 

The  angle  aoc  is  called  the  angle  of  diffraction 
The  ratio  ac :  ao  is  by  definition  the  sine  of  this  angle  ; 


DIFFRACTION   OF  LIGHT.  267 

hence  if  the   angle  be  measured,   the  ratio  can  be 
found  from  Table  3. 

^[130.  Determination  of  Angles  of  Diffraction. — 
An  ordinary  diffraction  grating  (see  §  101)  consists 
of  a  set  of  parallel  and  equidistant  lines  ruled  or 
photographed  on  glass  (Fig.  117).  A  candle-flame 
viewed  through  such  a  grating 
gives  several  images,  as  in  the 
case  of  a  netting  (Fig.  114)  ;  but 
these  images  are  all  in  a  single 
row  (Fig.  116).  The  relation 
between  the  wave-length,  dis- 
tance between  lines,  and  angle 
of  diffraction  is  the  same  as  in 

-C  IG.  1 1  /. 

the  case  of  a  netting  (^[  129). 
The  angle  of  diffraction  may  be  determined  either 
by  a  sextant  (^[  124),  or  by  a  spectrometer  (^[  126). 
In  any  case  the  lines  of  the  grating  must  be  per- 
pendicular to  the  graduated  arc  or  circle  by  which 
this  angle  is  to  be  determined. 

I.  A  coarse  diffraction  grating,  containing  from 
10  to  20  lines  to  the  millimetre,  is  to  be  mounted 
directly  in  front  of  the  tube  or  telescope  of  a  sextant 
(c,  Fig.  106,  ^[  121),  which  is  then  to  be  pointed 
at  a  distant  sodium  flame  (<f[  127).  When  the 
fixed  and  revolving  mirrors  are  nearly  parallel,  it 
should  be  possible  to  see  the  flame  (either  directly 
or  by  double  reflection)  with  at  least  two  images 
due  to  diffraction,  one  on  each  side  of  it  (see  a' a", 
Fig.  114).  The  experiment  consists  in  measuring 
the  angular  distance  between  the  two  side  images 


268  WAVE-LENGTHS.  [Exp.  47. 

next  the  flame,  by  the  method  already  explained 
in  If  124. 

Let  a  and  b  (Fig.  11G)  represent  the  direct  and 
doubly  reflected  images  of  the  flame.  The  revolving 
mirror  is  first  set  so  that  the  side  image  at  the  left  of 
6  coalesces  with  the  side  image  at  the  right  of  a,  to 
form  a  compound  image,  c,  as  in  the  figure.  Then 
the  image  (£")  at  the  right  of  b  is  made  in  the  same 
way  to  coalesce  with  the  side  image  (a')  at  the  left 
of  a.  The  two  readings  are  then  subtracted  algebra- 
ically, one  from  the  other,  and  the  result  is  divided 
by  2  (as  in  ^[  124),  to  find  the  angle  subtended  by  the 
side  images  (a'  and  a",  Fig.  114).  This  angle  must 
again  be  divided  by  2  to  find  the  angular  distance  of 
either  of  the  side  images  from  the  flame.  This  an- 
gular distance  evidently  corresponds  to  the  angle  aoc 
(Fig.  115),  and  is,  accordingly,  the  angle  (a)  of  dif- 
fraction in  question. 

Since  the  wave-length  of  sodium  light  is  .0000589, 
we  have,  substituting  this  value  in  formula  III.,  ^f  129, 
for  the  distance  (cT)  between  two  lines  of  the  grating. 

d  =  .0000589  I. 


A  grating,  thus  tested,  serves 
as  a  convenient  scale  by  which 
the  diameters  of  small  objects 
may  be  determined.  Such  a 
FIG.  118.  sca|e  jg  interesting,  because  it 

represents  the  nearest  approach  to  an  absolute  stand- 
ard of  length  (see  §  5). 


1  130,  II.]  DIFFRACTION   OF  LIGHT.  269 

II.  Instead  of  a  sextant,  a  spectrometer  may  be  em- 
ployed to  measure  angles  of  diffraction.  If  the  grating 
is  mounted  in  the  centre  of  the  graduated  circle  (Fig. 
118),  so  as  to  be  perpendicular  to  the  collimator,  ab, 
the  reading  of  the  telescope,^/,  when  set  upon  one 
of  the  side  images,  will  determine  the  angle  of  dif- 
fraction in  question.  It  is  not  very  easy,  however, 
to  make  the  grating  accurately  perpendicular  to  the 
collimator,  and  the  slightest  deviation  affects  the 
angle  of  diffraction.  A  grating,  like  a  prism  (see 
^[  127)  is  found  to  have  a  position  of  minimum  devi- 
ation, when  it  is  equally  inclined  to  the  direct  and 
diffracted  rays  (see  de,  Fig.  118).  This  position  may 
be  found  by  trial  in  the  same  way  as  with  a  prism. 

When  the  method  of  minimum  deviation  is  em- 
ployed, the  formulae  of  ^[  129  must  be  somewhat 
modified.1 

The  wave-lengths  contained  in  Table  41  were 
determined  by  a  method  essentially  the  same  as  the 
one  here  given. 

1  In  Fig.  115  each  ray  is  supposed  to  lose  one  wave-length  with 
respect  to  the  next  before  reaching  the  grating.  If,  however,  the 
grating  is  equally  inclined  to  the  incident  and  diffracted  ray,  the 
loss  must  be  half  a  wave-length  before,  and  half  a  wave-length  after 
reaching  the  grating;  that  is,  ef=  i  /.  The  angle  aoc  will  represent 
also  half  the  total  angle  of  diffraction ;  or  aoc  =  \  a.  If  d  is  the  dis- 
tance de  between  the  lines,  we  have,  substituting  sin  i  a  =  sin  aoc  for 
ac  ^  ao  (see  H  129),  and  multiplying  by  2, 
1—2  dm  I  a. 


270  WAVE-LENGTHS.  [Exp.  48. 


EXPERIMENT   LXVIII. 

INTERFERENCE   OP   SOUND. 

*[[  131.  Determination  of  the  "Wave-Length  of  a  Tun- 
ing-Fork by  the  Method  of  Interference.  —  I.  The  two 
ends  of  a  thick-sided  rubber  tube,  about  half  a  metre 
long,  and  with  an  internal  diameter  of 
at  least  5  mm.,  are  joined  together,  as 
in  Fig.  119,  by  a  Y-joint,  and  a  tube 
connected  with  the  stem  of  the  Y  is 
held  to  the  ear.  A  tuning-fork  making 
from  400  to  600  vibrations  per  second 
(as  for  instance  a  "violin  A-fork"  or  a 
"  C-fork  "  just  above  it)  is  then  touched 

lightly  to  the  tube  at  different  points,  as 
FIG.  119.        in   the   figure       The   ]lote   emitte(j   will 

generally  be  plainly  heard ;  but  two  or  more  points 
will  be  found  at  which  the  sound  is  nearly  extin- 
guished. These  points  are  to  be  marked  with  ink 
on  the  rubber  tube.  Then  the  tube  is  to  be  discon- 
nected from  the  Y-joint,  straightened  out,  but  not 
stretched,  and  the  distance  between  adjacent  marks 
carefully  determined  by  a  metre  rod. 

The  extinction  of  the  sound  is  due  to  the  inter- 
ference of  vibrations  reaching  the  Y-joint  by  the  two 
different  channels  (§  100),  which  differ  either  by 
half  a  wave-length,  or  by  some  odd  multiple  of  half 
a  wave-length.  It  follows  that  two  adjacent  points, 


1  131.]    •  INTERFERENCE   OF   SOUND.  271 

a  and  b  (Fig.  119),  where  the  sound  reaches  a  mini- 
mum, must  be  half  a  wave-length  apart.  To  find 
the  length  of  a  wave  of  sound  created  in  the 
tube  by  the  vibration  of  the  tuning-fork  in  question, 
we  have  therefore  only  to  multiply  the  distance  ab 
by  2. 

Wave-lengths  depend  more  or  less  upon  the  tem- 
perature of  the  air  in  the  tube,  which  should  there- 
fore be  noted.  They  are  generally  less  in  small  tubes 
than  in  the  open  air,  particularly  if  the  sides  of  the 
tube  be  yielding.  The  interference  is  never  complete, 
because  the  wave  which  travels  the 
longer  distance  becomes  weaker  than 
the  other,  and  hence  cannot  wholly 
destroy  it.  The  points  where  the  sound 
reaches  a  minimum  may  often  be  lo- 
cated more  exactly  when  a  fork  is  vi- 
brating feebly  than  when  it  is  sounding 
loudly. 

II.  In  place  of  a  rubber  tube,  we 
may  employ  a  pair  of  telescoping  U- 
tubes  (Fig.  120),  forming  a  closed  cir- 
cuit. Near  the  junctions  two  openings 
are  made.  One  of  these  is  connected 
with  the  ear,  the  other  receives  vibrations  propagated 
from  a  tuning-fork  through  the  air.  The  t\vo  chan- 
nels by  which  the  sound  reaches  the  ear  may  be  made 
unequal  in  length  by  drawing  out  the  tubes.  The 
difference  between  them  may  be  measured  by  gradu- 
ations on  the  inner  tube,  or  in  any  other  obvious 


272  WAVE-LENGTHS.  [Exp.  49. 

The  smallest  difference  between  the  two  channels 
which  can  produce  interference  is  half  a  wave-length  ; 
hence,  we  multiply  it  by  2  to  find  the 
wave-length  in  question. 

From  the  wave-length  of  a  fork  in 
air,  we  may  calculate  roughly  its  rate 
of  vibration  (^[  134,  formula  II.). 


EXPERIMENT   XLIX. 

RESONANCE. 

^[  132.  Determination  of  Wave-Lengths 
by  the  Method  of    Resonance.  —  A  me- 
tallic tube  or  "  resonator"  1|  metres 
long  and  10  cm.  in  diameter  (c,  Fig.  121) 
is  filled  with  water  ;  then  a  tuning-fork, 
making   from    200   to   300   vibrations 
per  second,  is  held  near  the  mouth  of 
the  tube,  while  the  water  escapes  by 
the  spout,  e.    When  the  water  falls  to 
a    certain   level,  the  note  emitted  by 
the  fork,  instead  of  dying  away,  will 
suddenly  swell  out.    The  flow  of  water 
FIG.  121.        is   then    checked.      Water    from    the 
faucet  is  now  admitted  to  the  resona- 
tor by  the  spout  e,  and  again  allowed  to  escape,  with 
a  view  to  finding  at  what  level  it  gives  the  maximum 
resonance.     The  variation    in   the   loudness    should 
be  observed  both  when  the  water  is  rising  and  when 
it  is  falling.    By  alternately  increasing  and  diminish- 


IT  133,  U.]  RESONANCE.  273 

ing  the  quantity  of  water  in  the  tube,  the  desired 
level  may  be  located  within  a  millimetre.  This  level 
is  then  read  by  the  gauge  ab,  consisting  of  a  milli- 
metre scale,  6,  and  a  glass  tube,  a,  connected  by  a 
rubber  tube  (cT)  with  the  resonator. 

The  fork  is  now  kept  in  vibration  while  the  level 
of  the  water  is  allowed  to  fall  to  a  much  greater  depth 
than  before.  A  second  point  of  resonance  is  thus  lo- 
cated in  the  same  way  as  the  first.  The  temperature 
of  the  air  within  the  tube  should  be  carefully  noted. 

The  distance  between  the  two  points  of  maximum 
resonance  is  found  by  subtracting  one  scale-reading 
from  the  other.  This  distance  is  (see  §  99)  exactly 
half  a  wave-length,  and  hence  must  be  multiplied  by 
2  to  find  the  wave-length  of  the  fork. 

The  rate  of  vibration  of  the  fork  may  now  be  cal- 
culated approximately,  as  will  be  explained  in  ^f  134, 
by  formula  II.  of  that  section. 


EXPERIMENT    L. 

MUSICAL    INTERVALS. 
1f  133.     Determination     of     Musical     Intervals.  —  I. 

METHOD  OF  INTERFERENCE.  — The  wave-lengths  of 
two  forks  are  to  be  determined  as  in  Experiment  48, 
taking  care  that  the  temperature  of  the  air  is  the 
same  in  both  cases,  and  the  musical  interval  between 
the  forks  is  to  be  calculated  as  in  ^[  134,  III. 

II.    METHOD  OF  RESONANCE.  —  Instead  of  using 
the  method  of  interference,  we  may  determine  the 
18 


274  WAVE-LENGTHS.  [Exp.  50. 

wave-lengths  of  two  forks  by  the  method  of  reso- 
nance, as  in  Experiment  49,  with  care  as  before  to 
avoid  changes  of  temperature.  The  musical  interval 
should  be  calculated  in  the  same  way  (^[  134,  III.). 

III.  PYTHAGOKEAN-  METHOD.  —  An  instrument 
which  will  be  found  convenient  for  the  determination 
of  musical  intervals  is  represented  in  Fig.  122.  It  is 


FIG.  122. 

called  the  "  monocJiord"  and  is  attributed  to  Pythag- 
oras. In  modern  instruments,  it  consists  of  a  steel 
wire,  fbcdeg  fastened  to  a  board  at  #,  then  passing 
over  two  "  bridges  "  (or  triangular  supports,  e  and  6) 
round  a  pulley  (/)  to  a  weight  (Ji)  by  which  it  is 
kept  stretched  with  a  constant  force.  The  positions 
of  the  bridges  are  determined  by  a  graduated  scale. 

The  wire  (be)  is  set  in  vibration  by  a  bow  (ai),  and 
the  distance  between  the  bridges  (b  and  e)  is  varied 
until  the  note  emitted  by  the  wire  is  in  unison  with 
one  of  the  forks.  The  distance  (be}  is  then  adjusted 
so  as  to  produce  unison  with  the  other  fork.  From 
the  two  distances  in  question,  the  interval  between 
the  forks  is  to  be  calculated  as  in  ^[  134  (For- 
mula III.). 

Determinations  with  a  monochord  should  be  at- 


1  133,  IV.]  HARMONICS.  275 

tempted  only  by  students  having  a  more  or  less 
musical  ear.  The  exact  adjustment  of  two  notes 
in  unison  may  be  inferred  from  the  cessation  of 
"  beats  "  (Exp.  53). 

IV.  HARMONIC  METHOD.  When  the  musical  inter- 
val between  two  forks  has  been  determined  by  any 
of  the  preceding  methods,  or  simply  recognized  by 
the  ear,  the  exactness  of  the  interval  in  question  may 
be  tested  as  follows :  The  bridges  b  and  e  (Fig.  122) 
are  first  placed  at  a  distance  which  is  the  least  com- 
mon multiple  of  the  two  distances  giving  unison  with 
the  two  forks.  By  touching  the  string  lightly  with  a 
feather  (e,  Fig.  122)  at  certain  points,  it  may  be 
made  to  vibrate  in  segments  as  in  the  figure.  The 
number  of  segments  is  first  made  such  that  the  string 
is  nearly  in  unison  with  one  of  the  two  forks,  and  the 
distance  (de)  adjusted  if  necessary  so  that  the  unison 
may  be  perfect.  If  the  wire  can  be  made  to  divide 
in  such  a  manner  as  to  sound  in  unison  with  the 
other  fork,  there  must  be  an  exact  musical  interval 
between  the  forks.  If,  on  the  other  hand,  beats  are 
heard,  the  interval  is  probably  inexact,  and  by  an 
amount  which  may  be  estimated  from  the  frequency 
of  the  beats  (Exp.  53). 

For  the  practical  application  of  this  method,  the 
monochord  should  be  capable  of  giving  a  very  low 
note,  at  least  two  octaves  flf  134)  below  the  lower 
fork  ;  hence  the  tension  of  the  wire  must  not  be  too 
great.  The  lowest  note  which  a  string  can  give  out 
under  given  circumstances  is  called  its  "  fundamen- 
tal tone,"  The  other  tones  are  caused  by  its  division 


276  MUSICAL   INTERVALS.  [Exp.  50. 

into  segments,  separated  by  still  points  or  "  nodes." 
These  tones  are  called  the  "• harmonics'"  of  the  string. 
The  musical  interval  between  any  two  harmonics 
may  be  calculated  from  the  number  of  vibrating 
segments  (see  ^|  134,  IV.),  which  must  therefore  be 
noted  in  each  case. 

*[[  134.  Theory  of  Musical  Intervals.  —  If  a  tuning- 
fork  gives  out  n  waves  each  I  centimetres  long  in  one 
second,  then  the  furthest  wave  must  be  nl  centi- 
metres off  from  the  fork  at  the  end  of  that  space 
of  time ;  and  since  it  travels  nl  cm.  in  1  sec.,  the 
velocity  of  sound  must  be  nl  cm.  per  sec.  The  fun- 
damental equation  connecting  the  number  (n)  of 
vibrations  per  second,  the  wave-length  (f),  and  the 
velocity  of  sound  (w)  is,  therefore, — 

v  =  nl.  I. 

The  velocity  of  sound  in  air  of  any  temperature 
may  be  found  from  Table  15  B.  If  the  humidity  is 
unknown,  a  mean  value  (60  per  cent)  may  be  as- 
sumed ;  then  if  the  wave-length  of  a  given  fork  is  £, 
we  have,  — 

« 

II. 


When  two  forks  give  n'  and  n"  vibrations  per  sec- 
ond, with  wave-lengths  respectively  of  V  and  I" 
centimetres,  we  have  from  II.,  — 

n'  =  v  +  V,  (1) 

and  n"  =  v  + 1" ;  (2) 

hence,  dividing  (1)  by  (2), 

n'  :  n"  ::  I"  :  I'.  III. 


H  134.]  MUSICAL   INTERVALS.  277 

The  ratio  of  the  rates  of  vibration  is  called  the 
musical  interval  between  the  forks,  and  is  accordingly 
in  the  inverse  ratio  of  their  wave-lengths. 

Formula  III.  is  applicable  to  a  wire  as  well  as  to  a 
tube.  When  a  wire  of  the  length  be  divides  into  N 
segments,  the  length  of  each  must  be  be  ~  N;  we 
have  accordingly  for  the  lengths  I '  and  I "  of  the  seg- 
ments formed  by  the  division  of  the  wire  (be)  into 
N'  and  N"  parts,  respectively,  — 

l'  =  be  +  N',  (3) 

l"  =  be  +  N";          (4) 
hence,  dividing  (4)  by  (3), 

I"  :  V  :  :  N'  :  N",        (5) 
which,  substituted  in  III.,  gives 

n'  :  n"  :  :  N>  :  N".  IV. 

This  shows  that  the  rates  of  vibration  of  different 
harmonics  are  proportional  to  the  number  of  vibrating 
segments  in  the  wire. 

It  has  been  stated  that  the  ratio  between  two  rates 
of  vibration,  n'  and  n",  determines  the  interval  be- 
tween the  two  notes  to  which  they  correspond.  The 
ordinary  musical  scale  consists  of  a  series  of  notes 
whose  rates  of  vibration,  whether  high  or  low,  are 
always  relatively  proportional  to  the  following  num- 
bers set  beneath  their  names :  — 

DO  RE  MI  FA  SOL  LA  SI  DO 

24          27          30          32  36          40          45          48 

The  interval  between  the  first  and  third  note  of  this 
series  is  called  a  "third;"  between  the  first  and 


278  MUSICAL  INTERVALS.  [Exp.  50. 

fourth,  a  "  fourth,"  etc.  The  first  two  are  said  to  be 
one  tone  apart ;  the  last  two,  one  semitone  apart. 
The  most  common  musical 'intervals  may  be  arranged 
as  follows,  according  to  the  simplicity  of  the  ratios 
which  they  involve  when  reduced  to  their  lowest 
terms :  — 

Name.                      Ratio.  Name.                      Ratio.  Name.                     Ratio. 

Unison    .     .     .     1:1  Fourth     .     .     .     4:3  Minor  Third  .6:5 

Octave    .     .     .    2:1  Sixtli  ....5:3  Whole  Tone  .9:8 

Fifth  ....3:2  Third  ....5:4  Semitone   .     .  16  :  15 

The  sum  of  two  or  more  intervals  is  always  repre- 
sented by  the  product  of  the  ratios  in  question  ;  thus, 
when  we  say  that  two  notes  are  an  octave  and  a  fifth 
apart,  we  mean  that  the  higher  makes  one  and  one 
half  times  as  many  vibrations  per  second  as  the 
octave  of  the  lower  note  ;  or,  again,  twice  as  many 
vibrations  as  a  note  a  "  fifth  "  above  the  lower  note  ; 
that  is,  in  either  case,  three  times  as  many  vibrations 
as  the  lower  ^ote  itself.  In  the  same  way  an  inter- 
val of  two  octaves  corresponds  to  the  ratio  4  :  1 
between  the  rates  of  vibration ;  an  interval  of  three 
octaves  corresponds  to  the  ratio  8  :  1,  etc.  It  is  a 
fact  to  be  noted  that  the  musical  intervals  involving 
the  simplest  ratios  are  the  most  agreeable  to  the  ear. 


END    OF   PART   FIRST. 


4098 


A 000938035    3 


UNIVEI 


LU  .KS 

UBRARY 


